Xenharmonic Reference - User contributions [en]

26edo

2026-05-31T04:18:00Z

Lériendil: /* JI approximation */

'''26edo''', or 26 equal divisions of the octave (sometimes called '''26-TET''' or '''26-tone equal temperament'''), is the equal tuning featuring steps of (1200/26) ~= 46.15 [[cent]]s, 26 of which stack to the perfect octave [[2/1]].

26edo has a [[perfect fifth]], 692.3{{c}}, which is tuned even flatter than that of [[19edo]]. Its [[5L 2s|diatonic]] scale is thus very [[soft]] ([[homoioheptatonic]]). Its thirds can still be taken, if inaccurately, to approximate [[6/5]] and [[5/4]], supporting [[Meantone]]. In terms of [[7-limit]] properties, 26edo is notably the smallest EDO to distinguish all of [[9/8]], [[8/7]], [[7/6]], 6/5, and 5/4 (although "9/8" in particular is far closer to [[10/9]]), and does so [[consistent]]ly.

Where 26edo truly shines, however, is in higher limits. We can observe it closely approximates both the [[7/4|7th]] and [[11/8|11th]] harmonics (to within half a cent for the former and 3 cents for the latter). Structurally, 26edo's fifth spans 15 edosteps, which means that it can be split into 3 parts and into 5. Both splits result in almost perfectly just intervals: 8/7 serves as 1/3 (5 edosteps), resulting in [[Slendric]] temperament, and [[13/12]] as 1/5 of the fifth (3 edosteps). {{adv|These intervals are tuned particularly well as a result of 26edo approximating the [[natave|natural]] fifth of ''e''2/5.}} Anchored by 8/7, 11/8, and 13/12, 26edo consistently represents the 13-[[odd-limit]], beating [[22edo]]'s consistency record of 11. Prime 17 can also be included in the mix, as it is tuned similarly to 13 and 3, although intervals of 15 are inconsistently mapped.

26edo overall, despite ostensibly supporting familiar harmonic organization in the form of Meantone, presents that organization vastly differently from [[12edo]] due to its flat tuning. Furthermore, 26edo includes [[13edo]] as a subset, and with it, the [[oneirotonic]] scale. And as a claimant for the smallest EDO to merit consideration as a [[17-limit]] system, and with primes 7 and 11 tuned far more accurately than 3 and 5, 26edo is quite capable of supporting harmonic systems that rely at most minimally on the diatonic scale or [[5-limit]] harmony at all.

== Theory ==
=== JI approximation ===
26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.

With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony|chthonic]] medials); and, as while 7 is accurate, 5 is flat enough in 26edo, [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.

26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.
{{Harmonics in ED|26|31|0}}

=== Edostep interpretations ===
26edo's edostep has the following interpretations in the 11-limit:
* 25/24 (the difference between 6/5 and 5/4)
* 33/32 (the difference between 4/3 and 11/8)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 64/63 (the difference between 9/8 and 8/7)
* 80/77 (the difference between 11/10 and 8/7)

=== Intervals and notation ===
Similar to [[19edo]], 26edo can be notated entirely with standard [[diatonic notation]], with #/b = 1\26 and x/bb = 2\26. The equivalences are Cx = Dbb, E# = Fbb, and Ex = Fb. JI approximations are given in the 17-limit 21-odd-limit, aside from the single edostep and its complement. Approximations within 3{{c}} are given in [brackets]. Harmonics 3-21 are bolded; inconsistent intervals (involving 15) are italicized.

{| class="wikitable"
|+
! Edostep !! Cents !! Notation !! 17-limit JI approximation !! ADIN interval category
|-
| 0
| 0
| C
| 1/1
| unison
|-
| 1
| 46.2
| C#
| 25/24, 33/32, [36/35], 49/48, 64/63
| superunison
|-
| 2
| 92.3
| Cx, Dbb
| ''15/14'', '''17/16''', 18/17, 21/20, 22/21
| farminor second
|-
| 3
| 138.5
| Db
| 12/11, [13/12], 14/13, ''16/15''
| (supra)minor second
|-
| 4
| 184.6
| D
| '''9/8''', [10/9], 11/10
| (sub)major second
|-
| 5
| 230.8
| D#
| [8/7], 15/13, 17/15
| farmajor second
|-
| 6
| 276.9
| Dx, Ebb
| 7/6, 13/11, 20/17
| farminor third
|-
| 7
| 323.1
| Eb
| 6/5, 17/14
| (supra)minor third
|-
| 8
| 369.2
| E
| '''5/4''', 11/9, 16/13, 21/17, [26/21]
| (sub)major third
|-
| 9
| 415.4
| E#, Fbb
| 9/7, [14/11]
| farmajor third
|-
| 10
| 461.5
| Ex, Fb
| 13/10, [17/13], '''21/16''', 22/17
| subfourth
|-
| 11
| 507.7
| F
| 4/3, ''15/11''
| perfect fourth
|-
| 12
| 553.8
| F#
| '''[11/8]''', 18/13
| (sub)augmented fourth
|-
| 13
| 600
| Fx, Gbb
| 7/5, 10/7, 17/12, 24/17
| tritone
|-
| 14
| 646.2
| Gb
| 13/9, [16/11]
| (supra)diminished fifth
|-
| 15
| 692.3
| G
| '''3/2''', ''22/15''
| perfect fifth
|-
| 16
| 738.5
| G#
| 17/11, 20/13, [26/17], 32/21
| superfifth
|-
| 17
| 784.6
| Gx, Abb
| [11/7], 14/9
| farminor sixth
|-
| 18
| 830.8
| Ab
| 8/5, '''13/8''', 18/11, [21/13], 34/21
| (supra)minor sixth
|-
| 19
| 876.9
| A
| 5/3, 28/17
| (sub)major sixth
|-
| 20
| 923.1
| A#
| 12/7, 17/10, 22/13
| farmajor sixth
|-
| 21
| 969.2
| Ax, Bbb
| '''[7/4]''', 26/15, 30/17
| farminor seventh
|-
| 22
| 1015.4
| Bb
| [9/5], 16/9, 20/11
| (supra)minor seventh
|-
| 23
| 1061.5
| B
| 11/6, 13/7, '''''15/8''''', [24/13]
| (sub)major seventh
|-
| 24
| 1107.7
| B#, Cbb
| 17/9, 21/11, ''28/15'', 32/17, 40/21
| farmajor seventh
|-
| 25
| 1153.8
| Bx, Cb
| [35/18], 48/25, 63/32, 64/33, 96/49
| suboctave
|-
| 26
| 1200
| C
| 2/1
| octave
|}

== Compositional theory ==
=== Tertian structure ===
{| class="wikitable"
|+Thirds in 26edo
!Quality
|Farminor
|'''Supraminor'''
|'''Submajor'''
|Farmajor
|-
!Cents
|276.9
|'''323.1'''
|'''369.2'''
|415.4
|-
!Just interpretation
|7/6 (+10.0{{c}})
|'''6/5 (+7.4{{c}})'''
|'''11/9 (+22.2{{c}}) 16/13 (+9.8{{c}}) 5/4 (-17.1{{c}})'''
|14/11 (-2.1{{c}}) 9/7 (-19.7{{c}})
|-
!Steps
|6
|'''7'''
|'''8'''
|9
|}

Diatonic thirds are bolded.

=== Chords ===
{{WIP}}

TODO:
* write about flattone

=== Scales ===
==== Organization of MOSes ====
26edo has six distinct intervals that define octave-periodic generator structures, not counting those of 13edo. These generator structures and consequent [[MOS]] scales organize themselves into two loops of three, each linked by the operation of tripling.
{| class="wikitable"
|+
!
! colspan="3" |Loop 1
|-
!Temperament
|[[Slendric]]
|[[Flattone]]
|[[Superkleismic]]
|-
!Scale (albitonic)
|5-5-5-5-6
|4-4-3-4-4-4-3
|2-5-2-5-2-5-5
|-
!Generator
|5\, 21\
|11\, 15\
|7\, 19\
|-
!
! colspan="3" |Loop 2
|-
!Temperament
|[[Quartonic]]
|[[Bleu]]
|[[Roman]]
|-
!Scale (albitonic)
| - (11-note scale has >10 steps interval)
|3-3-3-3-3-3-3-3-2
|7-1-7-1-1-7-1-1
|-
!Generator
|1\, 25\
|3\, 23\
|9\, 17\
|}
{{WIP}}

== Multiples ==
=== 104edo ===
104edo is a strong no-5 [[Parapyth]] tuning.
{{Harmonics in ED|104|47|0}}
=== 130edo ===
130edo adds 26edo's accurate 7/4 and [[10edo]]'s accurate 13/8 to [[65edo]], resulting in a strong 2.3.5.7.11.13.19.23.31.47 system. It is a good [[Hemiwurschmidt]] tuning. It is also useful as an example for interval categorization.
{{Harmonics in ED|130|47|0}}

{{navbox EDO}}
{{Cat|edos}}

26edo

2026-05-31T04:17:11Z

Lériendil: Undo revision 7291 by Hotcrystal0 (talk)

'''26edo''', or 26 equal divisions of the octave (sometimes called '''26-TET''' or '''26-tone equal temperament'''), is the equal tuning featuring steps of (1200/26) ~= 46.15 [[cent]]s, 26 of which stack to the perfect octave [[2/1]].

26edo has a [[perfect fifth]], 692.3{{c}}, which is tuned even flatter than that of [[19edo]]. Its [[5L 2s|diatonic]] scale is thus very [[soft]] ([[homoioheptatonic]]). Its thirds can still be taken, if inaccurately, to approximate [[6/5]] and [[5/4]], supporting [[Meantone]]. In terms of [[7-limit]] properties, 26edo is notably the smallest EDO to distinguish all of [[9/8]], [[8/7]], [[7/6]], 6/5, and 5/4 (although "9/8" in particular is far closer to [[10/9]]), and does so [[consistent]]ly.

Where 26edo truly shines, however, is in higher limits. We can observe it closely approximates both the [[7/4|7th]] and [[11/8|11th]] harmonics (to within half a cent for the former and 3 cents for the latter). Structurally, 26edo's fifth spans 15 edosteps, which means that it can be split into 3 parts and into 5. Both splits result in almost perfectly just intervals: 8/7 serves as 1/3 (5 edosteps), resulting in [[Slendric]] temperament, and [[13/12]] as 1/5 of the fifth (3 edosteps). {{adv|These intervals are tuned particularly well as a result of 26edo approximating the [[natave|natural]] fifth of ''e''2/5.}} Anchored by 8/7, 11/8, and 13/12, 26edo consistently represents the 13-[[odd-limit]], beating [[22edo]]'s consistency record of 11. Prime 17 can also be included in the mix, as it is tuned similarly to 13 and 3, although intervals of 15 are inconsistently mapped.

26edo overall, despite ostensibly supporting familiar harmonic organization in the form of Meantone, presents that organization vastly differently from [[12edo]] due to its flat tuning. Furthermore, 26edo includes [[13edo]] as a subset, and with it, the [[oneirotonic]] scale. And as a claimant for the smallest EDO to merit consideration as a [[17-limit]] system, and with primes 7 and 11 tuned far more accurately than 3 and 5, 26edo is quite capable of supporting harmonic systems that rely at most minimally on the diatonic scale or [[5-limit]] harmony at all.

== Theory ==
=== JI approximation ===
26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.

With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony|chthonic]] medials), and, as while 7 is accurate, 5 is flat enough in 26edo, [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.

26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.
{{Harmonics in ED|26|31|0}}

=== Edostep interpretations ===
26edo's edostep has the following interpretations in the 11-limit:
* 25/24 (the difference between 6/5 and 5/4)
* 33/32 (the difference between 4/3 and 11/8)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 64/63 (the difference between 9/8 and 8/7)
* 80/77 (the difference between 11/10 and 8/7)

=== Intervals and notation ===
Similar to [[19edo]], 26edo can be notated entirely with standard [[diatonic notation]], with #/b = 1\26 and x/bb = 2\26. The equivalences are Cx = Dbb, E# = Fbb, and Ex = Fb. JI approximations are given in the 17-limit 21-odd-limit, aside from the single edostep and its complement. Approximations within 3{{c}} are given in [brackets]. Harmonics 3-21 are bolded; inconsistent intervals (involving 15) are italicized.

{| class="wikitable"
|+
! Edostep !! Cents !! Notation !! 17-limit JI approximation !! ADIN interval category
|-
| 0
| 0
| C
| 1/1
| unison
|-
| 1
| 46.2
| C#
| 25/24, 33/32, [36/35], 49/48, 64/63
| superunison
|-
| 2
| 92.3
| Cx, Dbb
| ''15/14'', '''17/16''', 18/17, 21/20, 22/21
| farminor second
|-
| 3
| 138.5
| Db
| 12/11, [13/12], 14/13, ''16/15''
| (supra)minor second
|-
| 4
| 184.6
| D
| '''9/8''', [10/9], 11/10
| (sub)major second
|-
| 5
| 230.8
| D#
| [8/7], 15/13, 17/15
| farmajor second
|-
| 6
| 276.9
| Dx, Ebb
| 7/6, 13/11, 20/17
| farminor third
|-
| 7
| 323.1
| Eb
| 6/5, 17/14
| (supra)minor third
|-
| 8
| 369.2
| E
| '''5/4''', 11/9, 16/13, 21/17, [26/21]
| (sub)major third
|-
| 9
| 415.4
| E#, Fbb
| 9/7, [14/11]
| farmajor third
|-
| 10
| 461.5
| Ex, Fb
| 13/10, [17/13], '''21/16''', 22/17
| subfourth
|-
| 11
| 507.7
| F
| 4/3, ''15/11''
| perfect fourth
|-
| 12
| 553.8
| F#
| '''[11/8]''', 18/13
| (sub)augmented fourth
|-
| 13
| 600
| Fx, Gbb
| 7/5, 10/7, 17/12, 24/17
| tritone
|-
| 14
| 646.2
| Gb
| 13/9, [16/11]
| (supra)diminished fifth
|-
| 15
| 692.3
| G
| '''3/2''', ''22/15''
| perfect fifth
|-
| 16
| 738.5
| G#
| 17/11, 20/13, [26/17], 32/21
| superfifth
|-
| 17
| 784.6
| Gx, Abb
| [11/7], 14/9
| farminor sixth
|-
| 18
| 830.8
| Ab
| 8/5, '''13/8''', 18/11, [21/13], 34/21
| (supra)minor sixth
|-
| 19
| 876.9
| A
| 5/3, 28/17
| (sub)major sixth
|-
| 20
| 923.1
| A#
| 12/7, 17/10, 22/13
| farmajor sixth
|-
| 21
| 969.2
| Ax, Bbb
| '''[7/4]''', 26/15, 30/17
| farminor seventh
|-
| 22
| 1015.4
| Bb
| [9/5], 16/9, 20/11
| (supra)minor seventh
|-
| 23
| 1061.5
| B
| 11/6, 13/7, '''''15/8''''', [24/13]
| (sub)major seventh
|-
| 24
| 1107.7
| B#, Cbb
| 17/9, 21/11, ''28/15'', 32/17, 40/21
| farmajor seventh
|-
| 25
| 1153.8
| Bx, Cb
| [35/18], 48/25, 63/32, 64/33, 96/49
| suboctave
|-
| 26
| 1200
| C
| 2/1
| octave
|}

== Compositional theory ==
=== Tertian structure ===
{| class="wikitable"
|+Thirds in 26edo
!Quality
|Farminor
|'''Supraminor'''
|'''Submajor'''
|Farmajor
|-
!Cents
|276.9
|'''323.1'''
|'''369.2'''
|415.4
|-
!Just interpretation
|7/6 (+10.0{{c}})
|'''6/5 (+7.4{{c}})'''
|'''11/9 (+22.2{{c}}) 16/13 (+9.8{{c}}) 5/4 (-17.1{{c}})'''
|14/11 (-2.1{{c}}) 9/7 (-19.7{{c}})
|-
!Steps
|6
|'''7'''
|'''8'''
|9
|}

Diatonic thirds are bolded.

=== Chords ===
{{WIP}}

TODO:
* write about flattone

=== Scales ===
==== Organization of MOSes ====
26edo has six distinct intervals that define octave-periodic generator structures, not counting those of 13edo. These generator structures and consequent [[MOS]] scales organize themselves into two loops of three, each linked by the operation of tripling.
{| class="wikitable"
|+
!
! colspan="3" |Loop 1
|-
!Temperament
|[[Slendric]]
|[[Flattone]]
|[[Superkleismic]]
|-
!Scale (albitonic)
|5-5-5-5-6
|4-4-3-4-4-4-3
|2-5-2-5-2-5-5
|-
!Generator
|5\, 21\
|11\, 15\
|7\, 19\
|-
!
! colspan="3" |Loop 2
|-
!Temperament
|[[Quartonic]]
|[[Bleu]]
|[[Roman]]
|-
!Scale (albitonic)
| - (11-note scale has >10 steps interval)
|3-3-3-3-3-3-3-3-2
|7-1-7-1-1-7-1-1
|-
!Generator
|1\, 25\
|3\, 23\
|9\, 17\
|}
{{WIP}}

== Multiples ==
=== 104edo ===
104edo is a strong no-5 [[Parapyth]] tuning.
{{Harmonics in ED|104|47|0}}
=== 130edo ===
130edo adds 26edo's accurate 7/4 and [[10edo]]'s accurate 13/8 to [[65edo]], resulting in a strong 2.3.5.7.11.13.19.23.31.47 system. It is a good [[Hemiwurschmidt]] tuning. It is also useful as an example for interval categorization.
{{Harmonics in ED|130|47|0}}

{{navbox EDO}}
{{Cat|edos}}

Glossary

2026-05-26T22:05:54Z

Lériendil: /* Proper limit */

This page lists various terms conventionally used in xenharmony (or in some cases, general music theory as it applies to xen) that can be briefly described.

'''Don't put idiosyncratic terms here.''' When using personal terminology in an article, either explain it there or link to an article about your theory that explains the term.

== Basis ==
A '''basis''' (pl. ''bases'') for a [[#JI group|JI group]], or similar group, is a list of intervals called ''generators'' such that:
# anything in the group can be written as a stack of intervals of the basis or their inverses (possibly with repetition).
# the list is non-redundant in the sense that there is only one way to write any particular interval in the group as a stack of generators.

Examples:
* [2, 3/2, 5/4] is a basis for the [[5-limit]]; so is [2, 3, 5].
* [2, 5/3] and [2, 9, 5] are not bases for the 5-limit, on account of not satisfying condition 1.
* [2, 3, 5, 15] is not a basis for the 5-limit, on account of not satisfying condition 2.

JI groups are denoted using basis elements separated by full stops, for example 2.5.11/3.

Categories: RTT, JI, Math terms

== Binary ==
A '''binary''' scale is a scale with exactly two step sizes (usually denoted L and s). [[MOS]] scales are binary, but binary scales need not be MOS scales (e.g. melodic minor, LsLLLLs, is binary but not a MOS).

Categories: Scales

== Cent ==
A '''cent''' (abbreviated to c or ¢) is the conventional measurement unit of the logarithmic (perceptual) distance between [[Frequency|frequencies]]; in other words, the size of the [[interval]] between them. A cent is defined as a frequency ratio of 2^(1/1200), or a factor of about 1.0005778, such that the octave ([[2/1]]) spans exactly 1200 cents, and therefore that each step of [[12edo]] spans exactly 100.

Categories: Core knowledge

== Chiral ==
When a '''chiral''' scale has its step pattern reversed, it is no longer a mode of the original scale. A scale is '''achiral''' when this does not hold.

MOS scales and certain ternary scales such as [[blackdye]] are achiral, but many scales of interest such as [[zarlino]], [[diasem]] and [[Zil|Zil[14]]] are chiral.

The chiral pair of a chiral scale is conventionally denoted "right-hand", "RH", or "R", and "left-hand", "LH", or "L". An algorithm is usually used to determine the choice of L or R, but it's musically inconsequential. All you have to know is that one is L and one is R.

Categories: Scales

== Chord ==
A '''chord''' is a finite set of (usually three or more) pitches, often implying a context when the pitches are played together. Two chords are usually considered the same chord if they only differ by transposition.

Categories: Core knowledge

== Comma ==
'''Comma''' may refer to:
# a small JI interval. A fundamental fact about JI is that no stack of one prime is a stack of any other set of primes. Commas hence occur frequently in stacking-based JI.
# The commas of a regular temperament are the intervals it tempers out, {{adv|which can all be written as stacks of a certain number of commas known as the ''comma basis'' which suffice to determine every comma that is tempered out or every pair of intervals that is equated.}} "Tempering out" means that all JI ratios/stacks that are separated by that comma are equated, e.g. tempering out 81/80 not only equates 81/64 and 5/4 but also equates 40/27 and 3/2. This follows from the principles of regular temperament.
# An interval region of intervals around 20 cents, less than about 30 cents.

Categories: JI, RTT

== Complexity ==
The '''complexity''' of a rank-2 temperament is fairly easy to intuit: it is how many stacked generators are needed to reach simple JI ratios. There is often a tradeoff between simplicity and accuracy in temperaments. For example, 5-limit Schismic is a more accurate but more complex temperament than 5-limit Meantone, since more generators are needed to reach 5/4 in the former.

Categories: RTT

== Consistency ==
An approximation of an interval in an EDO (or otherwise an equal-step tuning) is '''consistent''' when it is both the closest direct approximation of the just interval available in the tuning, and the approximation regularly dictated by the [[#Val|val]] being used (whether [[#Patent val|patent]] or otherwise). Approximations where these two deviate from each other are correspondingly '''inconsistent'''.

For example, the interval [[7/6]] is inconsistent in [[34edo|34et]], since while [[7/4]] is defined as 27 steps and [[3/2]] as 20 steps (implying 7/6 to be 7 steps), 7/6 itself is slightly closer to 8 steps than to 7 steps of 34edo. {{Adv|In the alternative val 34d, where 7/4 is mapped to 28 steps instead, 7/6 becomes consistent but 7/4 itself is now inconsistent.}}

When discussing equal tunings, it is common to speak of the '''consistency limit''', the largest [[odd-limit]] in which every interval is mapped consistently, as well as the '''distinct consistency limit''', which adds the criterion that all intervals of the consistent odd-limit must be mapped to distinct intervals in the tuning. Such odd-limits can further be restricted to a particular [[#JI group|subgroup]] which the ET is considered as an approximation to.

Categories: EDO, JI

== Constant structure ==
A '''constant structure''' (CS; Erv Wilson's term) is a scale such that no two of its interval classes share a common interval.

Pythagorean diatonic is a constant structure:
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
|-
!1/1
|9/8
|81/64
|4/3
|3/2
|27/16
|243/128
|-
!9/8
|9/8
|32/27
|4/3
|3/2
|27/16
|16/9
|-
!81/64
|256/243
|32/27
|4/3
|3/2
|128/81
|16/9
|-
!4/3
|9/8
|81/64
|729/512
|3/2
|27/16
|243/128
|-
!3/2
|9/8
|81/64
|4/3
|3/2
|27/16
|16/9
|-
!27/16
|9/8
|32/27
|4/3
|3/2
|128/81
|16/9
|-
!243/128
|256/243
|32/27
|4/3
|1024/729
|128/81
|16/9
|}
But 12edo diatonic is not, because 600c is both a 3-step interval and a 4-step one:
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
|-
!0\12
|200.0
|400.0
|500.0
|700.0
|900.0
|1100.0
|-
!2\12
|200.0
|300.0
|500.0
|700.0
|900.0
|1000.0
|-
!4\12
|100.0
|300.0
|500.0
|700.0
|800.0
|1000.0
|-
!5\12
|200.0
|400.0
|class="thl"|600.0
|700.0
|900.0
|1100.0
|-
!7\12
|200.0
|400.0
|500.0
|700.0
|900.0
|1000.0
|-
!9\12
|200.0
|300.0
|500.0
|700.0
|800.0
|1000.0
|-
!11\12
|100.0
|300.0
|500.0
|class="thl|600.0
|800.0
|1000.0
|}

Some find CS a desirable property for JI scales, and some people find constant structure scales easier to navigate on keyboards.

A JI scale being a CS is ''not'' equivalent to it being a detempering of an equal temperament. The latter implies the former, but not vice versa.

Categories: Scales

== Detempering ==
'''Detempering''' a tempered scale results in a scale that has pitches in JI (or a temperament that tempers less). Each tempered pitch corresponds to one or more pitches in the detempered scale, which map to the tempered pitch under the temperament.

The Zarlino scale in 5-limit JI is a detempering of Meantone diatonic. Pental blackdye is another detempering of Meantone diatonic, but with some cases of multiple detempered pitches corresponding to a tempered pitch.

Categories: RTT, JI

== Enharmonic ==

=== Sense 1 ===
Two notes or intervals are enharmonic, or enharmonically equivalent, if they map to the same degree of the chromatic scale (the 12-note [[MOS]] scale generated by a [[perfect fifth]]). This can be generalized to pairs of notes separated by the difference between a chroma and a small step in a given scale, where enharmonic intervals are separated by a diesis, and can be equated by tempering out said diesis.

=== Sense 2 ===
A 17- or 19-note MOS scale generated by a perfect fifth, which assigns enharmonically equivalent diatonic intervals their own scale degrees by making the diatonic diesis a small scale step. Schismic[17] is usable as a scale for [[Schismic]] temperament.

=== Sense 3 ===
A Greek scale in which the lower two of the three intervals of a [[tetrachord]] are less than a semitone each.

=== Sense 4 (proscribed) ===
In [[12edo]], enharmonic notes in sense 1 are equated, which has led to a secondary use of "enharmonic" to refer to other equations between notes of a scaleform in some tuning system (such as B# = Cb in [[19edo]]). This particular use is discouraged due to the potential for confusion with other meanings of this already overloaded term.

== Equave ==
An '''equave''' or '''interval of equivalence''' is an interval that separates notes that are considered equivalent. Most commonly the octave (2/1), but 3/1, 3/2, and other intervals are sometimes used. See also [[Glossary#Non-Octave|Non-Octave]].

Categories: Core knowledge
== Extension, contorsion ==
An '''extension''' of a temperament is a temperament that interprets the tempered intervals of the original temperament within a larger [[Glossary#JI group|JI group]]. A '''weak extension''' introduces new tempered intervals in addition to those of the original temperament, whereas a '''strong extension''' uses the same set of intervals as the original temperament. The opposite of an extension is a '''restriction''', which interprets a temperament as a subset of the original JI group, and strong and weak restrictions are defined similarly.

For instance, [[Meantone]] introduces [[5-limit]] interpretations of intervals on a [[Chain of fifths|chain]] of tempered fifths by making the equivalence ([[3/2]])4 = [[Octave|2]]2 × 5/4 (tempering out the comma [[81/80]] and finding 5 at 4 fifths up). But if the chain of fifths is continued further, [[7-limit]] harmonies can be introduced: (3/2)2 × (5/4)2 = 2 × [[7/4]], which can be worked out to place 7 at 10 fifths up, a mapping of 7 known as ''septimal Meantone'', which is a strong extension of 5-limit Meantone.

Weak extensions are created by dividing the original period or (a choice of) generator into equal parts and then interpreting the split parts. As an example, Mothra is a temperament where the 3/2 Meantone generator is split into 3 parts, and then (3/2)^(1/3) is interpreted as [[8/7]]. It is a weak extension of pental Meantone, as Meantone natively doesn't have something that is one-third of a 3/2, to the 7-limit. Sometimes a weak extension may split the period instead of the generator; for example, Pajara (2.3.5.7[10 & 22]) is a weak extension of Archy (2.3.7[5 & 22]) that splits 2/1 into two 7/5's.

If you don't interpret the new intervals of a weak extension, the result is called '''contorsion'''. For example, if one were to take 2.3.5 Meantone and split the fifth into three equal parts without interpreting ~(3/2)1/3 as JI, the resulting temperament is a contorted 2.3.5 Meantone. If a rank-1 temperament is said to be contorted, it's still an instance of the concept of not fully interpreted weak extensions, but implies that the equal tuning is just a multiple of, and has the same cent value mappings as a subset equal tuning. For example, 36edo is contorted in the 5-limit because it uses the same cent value mappings as 12edo but adds uninterpreted intervals outside of 12edo. Note that the new generator 1\36 has no 5-limit interpretation. However, interpreting 1\36 as 64/63 interprets 36edo as a weak extension of 5-limit 12edo that adds prime 7.

Note that temperaments of different ranks are ''not'' considered extensions or restrictions of one another.

''For this wiki's guidelines on what extensions a given temperament name refers to, see [[Xenharmonic Reference:Guidelines]].''

Categories: RTT, Somewhat technical

== Generator ==
In xen theory, a '''generator''' is an interval that is stacked to obtain various intervals. Technically, ''generator'' has a number of slightly different senses:
* An element of a (chosen) [[glossary#Basis|basis]] for a [[glossary#JI_group|group]].
* For a rank-2 structure (temperament or [[MOS]]), a non-period generator (the choice of which is not unique). Examples:
** In [[Meantone]], which has period 1\1, the generator is (assuming pure 2/1) a flattened ~3/2 or a sharpened ~4/3. This corresponds to a generator of the MOS 5L2s (MOS diatonic).
** In [[Pajara]], which has period 1\2, there are 4 choices of octave-reduced generator: ~16/15, ~4/3, ~3/2, ~15/8. This is a generator of the MOS 2L8s.
* An element of a [[generator sequence]].

Categories: RTT, JI

== Hardness ==
The '''hardness''' of a [[binary scale]] (a scale with two distinct step sizes, ''L'' > ''s''), principally a [[MOS]], is the ratio ''L'':''s''. For instance, the diatonic ([[5L 2s]]) scale in [[17edo]] comprises the steps 3-3-1-3-3-3-1; its hardness is therefore 3:1. In [[34edo]], the diatonic is inherited from 17edo, and its step sizes are 6-6-2-6-6-6-2. 6:2 reduces down to 3:1.

Scales with a hardness greater than 2:1 (i.e. ''L'' > 2''s'') are called "hard", while scales with a hardness less than 2:1 (i.e. ''L'' < 2''s'') are called "soft". Binary scales with a hardness of exactly 2:1 are called "basic". A finer gradation of terms for hardnesses is provided by [[TAMNAMS]].

The concept of hardness can also be extended to [[ternary scale]]s and higher - for instance, [[22edo]] [[zarlino]], 3-4-2-4-3-4-2, has hardness 4:3:2.

Categories: Scales

== Harmonic mode ==
A harmonic segment of the form ''n''::2''n'', considered as an octave-equivalent scale. For example, mode 7 of the harmonic series is 7:8:9:10:11:12:13:14.

Categories: JI
== Harmonic segment ==
Any finite set of consecutive harmonics in the harmonic series. Can be denoted ''m''::''n''. For example, 5:6:7:8:9:10 is written 5::10.

Categories: JI
== Harmonic series ==
The infinite sequence of whole-number frequency multiples, called ''harmonics'', above a fundamental frequency. The harmonics of 110 Hz are:
* 1st harmonic (fundamental): 110 Hz
* 2nd harmonic: 220 Hz
* 3rd harmonic: 330 Hz
* 4th harmonic: 440 Hz
* 5th harmonic: 550 Hz
* 6th harmonic: 660 Hz
* ...
Every JI interval occurs in the harmonic series as the pitch difference between some pair of harmonics.

Differences in relative loudnesses of various harmonics above a note, as well as deviations from mathematically exact harmonics (called ''inharmonicity''), are perceived as different timbres of the same note.

Categories: JI, Core knowledge

== Interval class ==
An '''interval class''' or '''generic interval''' is the set of all intervals that occur as a given number of steps in a given scale. For example, the interval class of fifths (4-step intervals) in 12edo diatonic is {700c, 600c}. Sometimes called an '''ordinal''', because these are called ordinal numbers in conventional diatonic theory: "seconds", "thirds", etc. Other schemes such as [https://en.xen.wiki/w/TAMNAMS TAMNAMS] use a 0-indexing scheme: "1-step" for "seconds", "2-step" for "thirds", etc. See also [[#k-step]].

Categories: Scales

== JI group ==
A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

JI groups are denoted by generators, called ''basis elements'' (the standard mathematical term) or ''formal primes'' in this context, separated by full stops: for example, 2.3.5.7 denotes the [[7-limit|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group.

Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to. On XR, 2.b/a.c/a.d/a may be written {{nowrap|2.(a:b:c:d)}} for brevity, for example {{nowrap|2.(5:7:11:13)}} = 2.7/5.11/5.13/5 (note that this is the JI group generated by 2/1 and the intervals of 5:7:11:13).

Groups can be generalized to non-JI generators, for example 2.√6 (representing a chain of perfect hemififths), or 2.φ.

A regular temperament starts with a JI group and maps the group to a tempered group. For example, Meantone maps 2.3.5 to the group generated by tempered 2 and tempered 3/2.

{{adv|Mathematically, a '''group''' is a set with}}
* {{adv|a binary operation * (for all group elements ''g'' and ''h'', ''g'' * ''h'' is also an element of the group)}}
* {{adv|the binary operation * is associative (thus no parentheses are needed when writing the group operation on more than two elements)}}
* {{adv|an identity element: a unique element ''e'' such that {{nowrap|''g'' * ''e'' {{=}} ''e'' * ''g'' {{=}} ''g''}} for all ''g'' in the group}}
* {{adv|an inverse element for every element: every ''g'' corresponds to a unique element ''g''-1 such that {{nowrap|''g'' * ''g''-1 {{=}} ''g''-1 * ''g'' {{=}} ''e''}}}}
{{adv|A subgroup ''generated by'' a subset of a group is the group formed by iterating the binary operation on elements in the subset. Equivalently, it is the smallest subgroup of the larger group containing that subset.}}

{{Adv|Groups in xen theory are typically a much more specific type of groups, namely [[wikipedia:Free abelian group|free abelian groups]].}}

Categories: JI, RTT, Math terms

== Limit ==
In just intonation, '''limit''' most commonly has two distinct senses:
* The ''p''-'''[[prime-limit]]''' is the set of all JI ratios with primes up to ''p'' in their prime factorization. 3/2, 5/3, 7/4, and 49/36 are all in the 7-prime-limit, but 11/7 is not.
* The ''n''-'''[[odd-limit]]''' is a set of JI intervals with both numerator and denominator at most ''n'' after all factors of 2 are removed. Equivalently, it is the set of all intervals that appear in the harmonic series scale ''k'':(''k''+1):...:2''k'' (and all their octave equivalents), where ''k'' = ''n''/2 + 1/2. For example, the 15-odd-limit is the set of intervals that occur in the harmonic series scale 8:9:10:11:12:13:14:15:16; 21/16 is not in the 15-odd-limit.
The term "limit" without qualification today more commonly means prime-limit, though Harry Partch who coined the term ''limit'' originally meant odd-limit.

=== Proper limit ===
While a prime-limit encompasses all ratios up to a given prime, '''proper prime-limit''' classifies JI ratios based only based on the ''highest'' prime they contain in either the numerator or denominator. Equivalently, it is all of the intervals of a prime limit that are not found in a lower prime limit. This has been called '''harmonic class''', but this is discouraged because (a) it's a vague term and there are potentially many situations where intervals could be classified into "classes", and (b) people e.g. often informally use "7-limit" to denote the proper 7-limit.

The proper ''p''-prime limit contains ''only'' ratios for which ''p'' is the highest prime number found in their factorizations. For example:
* [[7/4]] is in the proper 7-limit because 7 is the highest prime in its factorization.
* [[5/4]] is in the proper 5-limit, not proper 7-limit, even though it's within the 7-limit.
* [[9/7]] is in the proper 7-limit because the highest prime is 7 (since {{nowrap| 9 {{=}} 32 }}).

Similarly, the '''proper ''n''-odd-limit''' is the set of all ''n''-odd-limit intervals that are in no lower odd-limits. For example, 7/4 is in the proper 7-odd-limit, and 9/7 is in the proper 9-odd-limit.

This distinction helps differentiate between intervals that merely fall within a limit versus those that specifically use a particular prime or odd. Unlike regular harmonic limits, proper harmonic limits are mutually exclusive categories.

Categories: JI

== Linearly independent ==
A set of vectors (such as a set of [[monzo]]s or a set of [[val]]s) is '''linearly independent''' if no vector in the set is redundant: no nonzero multiple of a vector can be written as a sum of multiples of other vectors. In the Xenharmonic Reference we will often shorten this to '''independent'''. In other sources the term ''co-unique'' may be used. {{Adv|This is technically <math>\mathbb{Z}</math>-linear independence; <math>\mathbb{Z}</math>-modules and abelian groups are the same concept.}}

Examples (for vals):
* {{val|12 19 28}} and {{val|19 30 44}} ([[12edo]] and [[19edo]] [[Glossary#Val|patent val]]s in the [[5-limit]]) are independent.
* {{val|12 19 28}}, {{val|19 30 44}}, and {{val|31 49 72}} are not independent, since the [[31edo]] val is a sum of the 12edo and 19edo patent vals. {{adv|We say that three vectors are ''collinear'' if they taken together are not linearly independent though any two of them are.}}
* {{val|24 38 96}} and {{val|36 57 84}} are not independent, since they share a common multiple.

Examples of where this concept shows up in RTT:
* Basis elements for any applicable group must be independent.
* Two ''independent'' vals (equal temperaments) determine a rank-2 temperament, three ''independent'' vals determine a rank-3 one, ...

Categories: RTT, Math terms, Somewhat technical

== Monzo ==
A '''monzo''' is a vector (list of coordinates) representing a JI ratio, whose coordinates are (usually) prime exponents. Also called an '''interval vector''' or a '''prime count vector'''.

Example: 81/80 = 3^4/(2^4 * 5^1) = 2^-4 * 3^4 * 5^-1 can be written in monzo form as {{monzo|-4 4 -1}}.

Categories: RTT
== Neji ==
A '''neji''' ("near-equal/equivalent JI") is a (possibly somewhat loose) JI approximation to a non-JI scale (often an edo), usually a subset of a chosen harmonic mode. The term was introduced by Zhea Erose.

Nejis are usually written as enumerated chords (i.e. written in the form a:b:...:z in ascending order): for example, the 12edo neji used in Zhea Erose's Eurybia is 22:23:25:26:28:30:31:33:35:37:39:42:44.

Categories: JI

== Non-octave ==
A tuning or temperament which does not have 2/1 is called non-octave. This includes JI subgroups that do not include 2, such as 3.5.7, as well as equal temperaments such as [[Bohlen-Pierce|13edt]] or Bohlen-Pierce. Tunings and temperaments that map a multiple of 2/1 (but not 2/1 itself), such as 41ed4 or 4.5.7, are also included.

== Period ==
'''Period''' has the following related but different senses:
* The smallest unit at which a given scale repeats — a fraction of the equave but not necessarily the equave itself.
** Example: Pentawood (5L5s, LsLsLsLsLs) has period 1\5 (240c).
* One of the generators of a regular temperament, specifically chosen to be a fraction of the equave (usually 2/1). (We make this choice for musical reasons, though a group mathematically doesn't have a distinguished element called the "period".)
** Example: The temperament Blackwood has period 1\5.
The two senses are related in that a multiperiod scale or equal division often supports a multiperiod temperament interpretation, and a multiperiod temperament requires an equal division that supports it to be divisible by some number (namely, the number of temperament-periods in the equave).

Categories: Scales, RTT

== Pitch class ==
Assuming an equave, two pitches or two intervals belong to the same '''pitch class''' if they are separated by a multiple of the equave. Pitch class space is a circle, whereas pitch space is a line.

Lattice diagrams of JI or tempered scales show the pitches in a pitch-class lattice, a lattice one dimension lower than the original JI group, where equave differences are ignored.

Categories: Core knowledge
== Rank ==
The term '''rank''' just means "dimensionality". The rank of a temperament is the dimension of the group of tempered JI ratios under that temperament. A temperament like [[Meantone]] has rank (dimension) 2 because any interval in Meantone can be written as a stack of some number of tempered octaves and some number of tempered fifths. Any [[equal tuning]] is rank 1 because all intervals in an equal tuning are a stack of that tuning's step size.

Rank-2 temperaments deserve special mention as they can be described as stacking a single generator against a [[Glossary#Period|period]]. As a result, a very clear method for constructing scales from rank-2 temperaments exists, that being forming a [[MOS]] from the temperament's generator and period, which is quite nontrivial to generalize to systems of higher rank.

Categories: RTT, Math terms

== Regular temperament ==
:''Main article: [[Regular temperament]]''

A '''regular temperament''' (often just '''temperament''') is a way of assigning JI interpretations (from a chosen JI group) to intervals in a non-JI tuning. We assign the interpretations so that the stack of two JI ratios gets assigned to the stack of the corresponding tempered versions of the two ratios. We also assume that each JI ratio is assigned to one and only one cent value, unlike in irregular/well temperaments.

If you know what notes of a tempered tuning the ''basis generators'' of a chosen JI group get assigned to, that suffices to determine the interpretations assigned to any particular interval {{adv|(provided that every interval is indeed interpreted, as in the overwhelming majority of practical cases).}} This is how vals and mappings for regular temperaments work — they specify what tempered notes correspond to the basis elements of the JI group.

The study of regular temperaments is called [[regular temperament theory]] (RTT).

Categories: RTT

== Scale ==
A '''scale''' is a collection of pitches; two scales are considered the same scale if they only differ by transposition. Unlike chords, scales are usually ''periodic'', i.e. the same pattern of intervals repeats at some interval called the ''equave''. On the Xenharmonic Reference, ''scales are periodic or repeating unless stated otherwise.'' (Though of course, finite but nonrepeating pitch material may be useful to consider in some contexts like voicing and register, especially in harmonic series or spectralist music.) A scale can be visualized as a set of points in the circle of equave-equivalent pitch classes.

Categories: Core knowledge, Scale

== Signature ==
A '''signature''' is a list of numbers giving useful but incomplete information about an object. Usually refers to one of:
* a ''step signature'', a list of how many of each step size a scale has; e.g. 4L3m2s.
* a ''[[delta signature]]'', a list of frequency increases between adjacent notes measured relative to a reference frequency increase, e.g. +1+1+2 for the chord 6.465:7.465:8.465:10.465.

Categories: Somewhat technical
== ''k''-step ==
An abbreviation for "''k''-step interval". For example, the fifth in the diatonic scale is a 4-step. See also [[#Interval class]].

Categories: Scales

== Square-superparticular ==
A '''square-superparticular''' or '''square-particular''' is a superparticular of the form

<math>\frac{k^2}{k^2-1} = \frac{k}{k-1}\frac{k}{k+1} = \frac{\frac{k}{k-1}}{\frac{k+1}{k}},</math>

denoted S''k'' or S(''k'') in xen math.

A square-superparticular is the difference between consecutive suparparticulars. When a square-superparticular S''k'' is tempered out, it makes harmonics {{nowrap|''k'' - 1}}, ''k'', and {{nowrap|''k'' + 1}} equally spaced. For example, tempering out S9 = 81/80 makes harmonics 8, 9, and 10 equally spaced. Factoring a comma into a product of square-particulars, called an '''S-expression''', is often helpful for understanding it.

{{Adv|The ratio between two consecutive square-superparticulars is called an ''ultraparticular'', which has the form S''k''/S(''k'' + 1). Tempering out an ultraparticular equates the differences between three consecutive superparticulars.}}

Categories: RTT

== Superparticular ==
A '''superparticular''' or Delta-1 ratio is a ratio between two whole numbers which differ by 1: e.g. [[2/1]], [[3/2]], [[4/3]], [[5/4]], etc, representing intervals between consecutive members of the [[#Harmonic series|harmonic series]]. These are distinguished from '''superpartient''' ratios (all other rational ratios), which can be classified as Delta-2, Delta-3, etc. by the difference between their numerator and denominator. Note that the [[Glossary#Square-superparticular|ratio between consecutive superparticulars]] is itself superparticular.

Categories: JI, Math terms

== Ternary ==
A '''ternary''' scale is a scale with exactly three step sizes (usually denoted L, m, and s).

Categories: Scales

== Tertian ==
In standard music theory, '''tertian''' harmony refers to harmony where thirds are privileged as the main component of chords. The most basic tertian chords are root-third-fifth triads and their inversions, but larger chords such as dom7 (stacked M3-m3-m3) and major 11th (stacked M3-m3-M3-m3-M3) are often also considered tertian.

Categories: Chords

== Val ==
A '''val''' (short for "valuation") is a vector whose coordinates are step mappings of primes in an [[equal temperament]]. {{Adv|It can mathematically be called a "covector", since it is a kind of a vector "dual" (complementary) to interval vectors.}}

Example: 12et maps 2/1 to 12 steps, 3/1 to 19 steps (reduced: 7 steps), and 5/1 to 28 steps (reduced: 4 steps). We write this in val form as {{val|12 19 28}}. Vals can be ''evaluated'' at monzos (showing how the equal temperament maps the JI ratio) by multiplying each pair of corresponding entries and summing the results together. This can be seen as, for a monzo with entries m and a val with entries v, "stepping" by each v m times for its corresponding m. {{Adv|In linear algebra, this operation is called the dot product.}} This is denoted by {{val|val}}{{monzo|monzo}}. Evaluating this val at {{monzo|-4 4 -1}} (the monzo for 81/80) shows that 12et tempers out 81/80:

{{val|12 19 28}}{{monzo|-4 4 -1}} = 12 * -4 + 19 * 4 + 28 * -1 = -48 + 76 - 28 = 0.

Categories: RTT

=== Patent val ===
'''Patent vals''' are the most common kinds of vals to consider. The "patent" means that the closest approximations in the edo tuning in question are used for the step mappings. The above val is the 12edo patent val in the 5-limit. An example of a non-patent val is {{val|12 19 27}}, since the closest approximation to 5/1 in 12edo is not 27 steps, but 28 steps.

{{Adv|The concept of a patent val can be extended into the notion of a '''generalized patent val''' (GPV), or a "uniform map" in some sources. A GPV is essentially a patent val corresponding to an equal-step tuning that might not necessarily divide an exact 2/1. For instance, the val {{val|17 27 40}} is a GPV, as this consists of the closest approximations of primes 2, 3, and 5 in 17.1edo, where 5 differs from the approximation in 17edo proper (which is 39 steps).}}

Categories: RTT

== Variety ==
'''Variety''' (or '''interval variety''') refers to how many interval sizes an [[Glossary#Interval class|interval class]] comes in. We often refer to
* '''maximum variety''' (MV) if all varieties satisfy a certain bound and there is some variety equal to the bound (thus MV2 scales are ''not'' MV3)
* '''strict variety''' (SV) if all varieties (except equave multiples) are equal to some value.
For example, [[MOS]] scales can be defined as scales that are MV2. Mosdiatonic is SV2.

Categories: Scales

== Virtual fundamental ==
The '''virtual fundamental''' or '''missing fundamental''' associated with the pitches of a JI chord is the perception of a common frequency such that all of the notes of the chord are overtones of it. Humans perceive a virtual fundamental when they hear a (complete enough) set of overtones with the fundamental missing (though it's not as simple as just matching harmonics to a template; see [[Delta-rational chord]]). For example, the virtual fundamental of {220 Hz, 330 Hz, 440 Hz, 550 Hz, 660 Hz} is 110 Hz, so when a 2:3:4:5:6 chord is played on 220 Hz, you may hear a pitch at 110 Hz.

Categories: JI, Acoustics

Adaptive diatonic interval names

2026-05-26T03:00:50Z

Lériendil: italicized links that are redirects

The system of '''adaptive diatonic interval names (ADIN)''', developed by Vector and Leriendil, is a way to (mostly) uniquely label the intervals in an EDO based on size and relation to that EDO's patent fifth. It is ''diatonic'' because it attempts to behave predictably relative to MOSdiatonic staff notation, and it is ''adaptive'' because the differing qualities of diatonic intervals in different tunings are reflected in the interval names (that is to say, it "adapts" to different diatonic tunings). Finally, it is an ''interval naming system'', not a notation system, because it provides no way to write notes and labels intervals based on "what they are", not "what they do". (The creator of the ADIN system endorses [[Modified ups and downs notation|ups and downs notation]] for the latter.)

It is an attempt at formalizing the systems of interval qualities used by various xenharmonic resources on the internet.

=== On "major" vs. "supermajor" ===
A large number of resources, including Unque's theory page on xen.wiki, the Lumatone introductory video to 22edo, and the 31et.com page on 22edo, alongside resources for temperaments such as 41edo (31et.com) and 27edo (Lumatone), reserve "major/minor" for 5/4 and 6/5, distinguishing an unmarked "major" from terms like "supermajor" (and "minor" from terms like "subminor"). There is no obvious reason to do this other than 5-limit preferentialism, and it leads to ambiguity where "major" could either refer to specifically classical major intervals or more generally to any major interval. This is especially egregious as in [[Archy|archytas]] temperaments (a major form of structural temperament for the 7-limit), diatonic notation has the opposite behavior, leaving 9/7 and 7/6 unmarked while 5/4 and 6/5 get the extra prefix, meanwhile standard diatonic notation in just intonation, schismic, hemifamity, etc. uses "major" to refer to neither, instead denoting an interval in between the two, leaving only meantone temperaments unambiguous.

While the use of "major" in standard diatonic notation is not a problem on its own, it leads to a large degree of ambiguity with naming schemes wherein 5/4 is prioritized outside of meantone temperaments. To resolve this, ADIN provides the original label "nearmajor" for intervals with a similar quality to 5/4, and "nearminor" for intervals with a similar quality to 6/5. Additionally, "farmajor" and "farminor" are used to refer to intervals in the standard JI diatonic range, regardless of the actual tuning of the diatonic scale. Nearminor and nearmajor intervals may otherwise be called "classic(al)", "pental", or "ptolemaic" minor/major, which are terms used to describe the simple 5-limit intervals to which they correspond.

Unqualified "major" may refer to the range of major qualities collectively in cases like "either major key" or "the major thirds". However, when used to refer to a specific interval ("the major third"), it should refer to specifically the MOS diatonic intervals. Likewise for minor.

== Premise ==
ADIN names qualities, and then applies those names to intervals based on their distance from the nearest (possibly imaginary) diatonic neutral interval. The diatonic neutral intervals are as follows:

* Semidiminished unison (-3.5 fifths)
* Neutral second (-1.5 fifths)
* Neutral third (+0.5 fifths)
* Semiaugmented fourth (+2.5 fifths)
* Semidiminished fifth (-2.5 fifths)
* Neutral sixth (-0.5 fifths)
* Neutral seventh (+1.5 fifths)
* Semiaugmented octave (+3.5 fifths)

Intervals are named on a per-octave basis (that is, by octave-reducing, naming the interval, and adding back octaves according to conventional interval arithmetic), so the semidiminished unison and semiaugmented octave (which are lesser than and greater than the unison and octave respectively) do not actually appear in any interval names. Instead, they are chosen to ensure that the boundary between "unison" and "second" always falls precisely halfway between the perfect unison and the minor second.

These intervals may not exist in an edo (for instance, if it maps the fifth to an odd number of steps). This is okay, as they are being used as points of reference to compare to, not as actual necessary steps in the edo.

== Interval regions ==
Each neutral interval defines a series of regions (or "qualities") extending outwards from it, which are defined in terms of equal divisions of [[15/14]]. The use of 15/14 was proposed by [[User:Lériendil|Lériendil]] for threefold reasons:
* Firstly, 15/14 is a mapping of the [[apotome]] in [[aberschismic]] tunings: that is, it is the interval between [[7/6]] and [[5/4]] and between [[6/5]] and [[9/7]], and therefore the interval between the midpoint of 7/6 and 6/5, and the midpoint of 6/5 and 9/7;
* Secondly, it is close to 120 cents, which is the maximum amount of separation an interval can have from a diatonic neutral (assuming the fifth does indeed generate a diatonic scale), ensuring all intervals can be named;
* Finally, it is not itself an equal division of the octave, ensuring that no EDO intervals (aside from the true neutrals) land on region boundaries.
{| class="wikitable"
|+
!\25ed(15/14)
!Cents
!Major
!Minor
|-
|0
|0
| colspan="2" |neutral
|-
|0-2
|0-9.6
|tendoneutral
|artoneutral
|-
|2-5
|9.6-23.9
|submajor
|supraminor
|-
|5-10
|23.9-47.8
|nearmajor
|nearminor
|-
|10-15
|47.8-71.7
|farmajor
|farminor
|-
|15-20
|71.7-95.6
|supermajor
|subminor
|-
|20+
|95.6+
|ultramajor
|inframinor
|}
For instance, assuming a fifth is tuned to JI, the categories of thirds are found at <255c (inframinor), 256-279c (subminor), 280-303c (farminor), 304-327c (nearminor), 327-341c (supraminor), 342-360c (neutral, arto/tendo-), 361-375c (submajor), 376-398c (nearmajor), 399-422c (farmajor), 423-446c (supermajor), and >446c (ultramajor).

With these, the complete sets of intervals of each edo may be given a name. When an interval is an equal distance from two neutrals, thirds are always given precedence over fourths (so that an interval equidistant between the neutral third and neutral fourth is always a kind of third), and over seconds, which take precedence over unisons (except for the perfect unison and octave). The same rules apply to the complementary region of the octave. Fourths always take precedence below the tritone, and fifths always take precedence above it.

The exception is when the diatonic intervals coincide, in which case the conflated interval belongs to the category corresponding to its simplest diatonic interpretation (i.e. 240c is a second, not a third, and 480c is a fourth, not a third or (diminished) fifth). The same applies to oneirotonic and antidiatonic structures.

If there is only one kind of major or minor, drop all prefixes on major and minor. For example, if the only interval qualities found are "farminor", "neutral", and "farmajor", then rename "farminor" to "minor" and "farmajor" to "major". As a result, skip step 3.

== Disambiguation ==
In large edos, multiple intervals may be assigned the same name at the current point. This is where the disambiguation scheme comes into play. Based on the number of intervals in each category, a fixed set of names is assigned in order of size.
{| class="wikitable"
|+
!Quality
!2
!3
|-
|inframinor
|arto, inframinor
|
|-
|subminor
|sensaminor, gothminor
|sensaminor, septiminor, gothminor
|-
|farminor
|neominor, novaminor
|neominor, triminor, novaminor
|-
|nearminor
|valaminor, magiminor
|valaminor, pentaminor, magiminor
|-
|supraminor
|daemominor, aurominor
|
|-
|artoneutral
|subneutral, artoneutral
|
|-
|tendoneutral
|tendoneutral, supraneutral
|
|-
|submajor
|auromajor, daemomajor
|
|-
|nearmajor
|magimajor, valamajor
|magimajor, pentamajor, valamajor
|-
|farmajor
|novamajor, neomajor
|novamajor, trimajor, neomajor
|-
|supermajor
|gothmajor*, sensamajor
|gothmajor*, septimajor, sensamajor
|-
|ultramajor
|ultramajor, tendo
|
|}

In cases where there are two intervals belonging to the nearminor/major, farminor/major, and subminor/supermajor qualities, "pentamajor", "trimajor", and "septimajor" are substituted in for major thirds within 4.78{{c}} (1\25ed(15/14)) of the characteristic just intervals 5/4, 19/15, and 9/7 respectively**. If any major third acquires one of these subqualities, it is then propagated to its complement and other interval degrees.

=== Alternative system ===
Primarily in the case of tuning systems other than EDOs, or large EDOs where more than 3 intervals exist within the space of a single quality band, another fallback system can be used to assign subqualities to specific intervals.

"Trimajor" is defined as a radius of 1\25ed(15/14) around 19/15**, the same way as it is above. "Septimajor" then directly occupies the band 1\5ed(15/14) sharp of trimajor, while "pentamajor" occupies the band 9\50ed(15/14) flat of trimajor. The sharp edge of pentamajor is then taken to be the edge between auromajor and daemomajor. Subneutral and supraneutral intervals are not distinguished in this system.

Pentamajor and septimajor can variantly be defined to center around 5/4 and 9/7 as above, for the sake of consistency with the system generally employed for EDOs.

In cent values, with a justly tuned 3/2, the subqualities sharpward of the neutral third are then bounded as follows:
* 350.978 <- tendoneutral -> 360.533 <- auromajor -> 368.634 <- daemomajor -> 374.866
* 374.866 <- magimajor -> 382.967 <- pentamajor -> 392.522 <- valamajor -> 398.755
* 398.755 <- novamajor -> 404.467 <- trimajor -> 414.022 <- neomajor -> 422.643
* 422.643 <- shrubmajor* -> 428.355 <- septimajor -> 437.911 <- sensamajor -> 446.532

In that case, two intervals falling within the same subquality can then be disambiguated as "small" and "large", or three as "small", "mid", and "large".

* "Shrub-" can be replaced with "goth-".

** A variation would be for 5/4, 19/15, and 9/7 to be substituted here with sqrt(25/24), sqrt(722/675), and sqrt(54/49) above the neutral third, snapping all subqualities to the same positions relative the neutral third.

== Final steps ==
There are some additional replacements to be done:

1) Examine the diatonic fourth and whether it is major or minor. Remove the corresponding quality from all fourth names (for example, if the diatonic fourth is a farminor fourth, replace all instances of "minor fourth" with simply "fourth". Rename the opposing quality from "major" to "augmented", or "minor" to "diminished". If the fourth is any kind of neutral, no change is necessary to any interval names.

2) Label the diatonic fourth "perfect fourth" regardless of its quality.

3) Repeat for the unison, fifth, and octave.

3a) The result may create ambiguities with terms like "far octave" in some edos (the smallest edo to feature this problem being 26edo, between 25\26 and 27\26). In that case, restore "major" to octaves, fifteenths, etc above their perfect counterparts and which have ambiguous labels, and "minor" to fifteenths and above.

4) If quality is not necessary to distinguish intervals at all, remove it entirely (i.e. if there are only neutral intervals, do not specify "neutral").

== Qualities in small diatonic EDOs ==
Below lists the palettes of neutral and major qualities (noting that minor qualities always exist as the complements of major qualities) that can be found in diatonic EDOs below about 60, that is, the EDOs that do not require the disambiguation step. A few EDOs have two diatonic fifths, one which is divisible in two and one which is not. Both fifths are kept track of, but non-patent fifths are in parentheses.

Ultramajor qualities are treated separately, since they are ambiguous in degree. However, for EDOs with flat fifths ([[19edo]] or flatter) and which divide the perfect fourth in two, subminor and supermajor qualities are in fact interordinal (e.g. supermajor thirds are the same as sub(minor) fourths). These EDOs will be marked with an asterisk. Some EDOs with sharp fifths have ultramajor (and inframinor) intervals which are, however, not interordinal; these will be marked with a superscript plus sign.

=== Without a neutral third ===
EDOs without a neutral third have:
* with a step size 21.25-27.3{{c}} -> '''submajor, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[46edo|46]], [[47edo|47]]*, [[49edo|49]], [[50edo|50]], [[53edo|53]], [[56edo|56]]+ ([[52edo|52b]]+, [[54edo|54b]])

* with a step size 27.3-28.65{{c}} -> '''submajor, nearmajor, farmajor'''
** diatonic fifths: [[42edo|42]]+, [[43edo|43]]

* with a step size 28.65-31.85{{c}} -> '''submajor, nearmajor, supermajor'''
** diatonic fifths: ''[[39edo|39]]'', [[40edo|40]]

* with a step size 31.85-38.2{{c}} -> '''submajor, farmajor, supermajor'''
** diatonic fifths: [[32edo|32]], [[33edo|33]]*, ''[[36edo|36]]''

* with a step size 38.2-47.8{{c}} -> '''submajor, farmajor'''
** diatonic fifths: [[26edo|26]], [[29edo|29]]

* with a step size 47.8-63.7{{c}} -> '''nearmajor, supermajor'''
** diatonic fifths: [[19edo|19]]*, [[22edo|22]]

* with a step size > 63.7{{c}} -> '''major'''
** diatonic fifths: [[12edo|12]]

=== With a neutral third ===
EDOs with a neutral third have:
* with a step size 19.1-23.9c -> '''neutral, submajor, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[51edo|51]], [[52edo|52]]*, [[54edo|54]], [[55edo|55]], ''[[58edo|58]]'', [[61edo|61]]+, [[62edo|62]] (''[[57edo|57b]]''+, [[59edo|59b]])

* with a step size 23.9-31.85c -> '''neutral, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[38edo|38]]*, [[41edo|41]], [[44edo|44]], [[45edo|45]], [[48edo|48]] ([[47edo|47b]]+)

* with a step size 31.85-35.85c -> '''neutral, nearmajor, farmajor'''
** diatonic fifths: [[34edo|34]], [[37edo|37]]+

* with a step size 35.85-47.8c -> '''neutral, nearmajor, supermajor'''
** diatonic fifths: [[27edo|27]], [[31edo|31]]

* with a step size 47.8-71.65c -> '''neutral, (far)major'''
** diatonic fifths: [[17edo|17]], [[24edo|24]]

== Notes ==
The first EDO this system fails to name the intervals for is currently 159edo, as it has four intervals within each supermajor range.

== Extensions ==

=== Oneirotonic ===
Add an extra ordinal for "tritone" rather than just treating it as a special case for even edos. The chroma is the moschroma of oneirotonic.

=== Antidiatonic ===
The chroma is the moschroma of antidiatonic. Note that the pythagorean semidiminished unison is still the center of the unison range, despite being larger than 0c.

Adaptive diatonic interval names

2026-05-26T02:59:47Z

Lériendil: italiciz

The system of '''adaptive diatonic interval names (ADIN)''', developed by Vector and Leriendil, is a way to (mostly) uniquely label the intervals in an EDO based on size and relation to that EDO's patent fifth. It is ''diatonic'' because it attempts to behave predictably relative to MOSdiatonic staff notation, and it is ''adaptive'' because the differing qualities of diatonic intervals in different tunings are reflected in the interval names (that is to say, it "adapts" to different diatonic tunings). Finally, it is an ''interval naming system'', not a notation system, because it provides no way to write notes and labels intervals based on "what they are", not "what they do". (The creator of the ADIN system endorses [[Modified ups and downs notation|ups and downs notation]] for the latter.)

It is an attempt at formalizing the systems of interval qualities used by various xenharmonic resources on the internet.

=== On "major" vs. "supermajor" ===
A large number of resources, including Unque's theory page on xen.wiki, the Lumatone introductory video to 22edo, and the 31et.com page on 22edo, alongside resources for temperaments such as 41edo (31et.com) and 27edo (Lumatone), reserve "major/minor" for 5/4 and 6/5, distinguishing an unmarked "major" from terms like "supermajor" (and "minor" from terms like "subminor"). There is no obvious reason to do this other than 5-limit preferentialism, and it leads to ambiguity where "major" could either refer to specifically classical major intervals or more generally to any major interval. This is especially egregious as in [[Archy|archytas]] temperaments (a major form of structural temperament for the 7-limit), diatonic notation has the opposite behavior, leaving 9/7 and 7/6 unmarked while 5/4 and 6/5 get the extra prefix, meanwhile standard diatonic notation in just intonation, schismic, hemifamity, etc. uses "major" to refer to neither, instead denoting an interval in between the two, leaving only meantone temperaments unambiguous.

While the use of "major" in standard diatonic notation is not a problem on its own, it leads to a large degree of ambiguity with naming schemes wherein 5/4 is prioritized outside of meantone temperaments. To resolve this, ADIN provides the original label "nearmajor" for intervals with a similar quality to 5/4, and "nearminor" for intervals with a similar quality to 6/5. Additionally, "farmajor" and "farminor" are used to refer to intervals in the standard JI diatonic range, regardless of the actual tuning of the diatonic scale. Nearminor and nearmajor intervals may otherwise be called "classic(al)", "pental", or "ptolemaic" minor/major, which are terms used to describe the simple 5-limit intervals to which they correspond.

Unqualified "major" may refer to the range of major qualities collectively in cases like "either major key" or "the major thirds". However, when used to refer to a specific interval ("the major third"), it should refer to specifically the MOS diatonic intervals. Likewise for minor.

== Premise ==
ADIN names qualities, and then applies those names to intervals based on their distance from the nearest (possibly imaginary) diatonic neutral interval. The diatonic neutral intervals are as follows:

* Semidiminished unison (-3.5 fifths)
* Neutral second (-1.5 fifths)
* Neutral third (+0.5 fifths)
* Semiaugmented fourth (+2.5 fifths)
* Semidiminished fifth (-2.5 fifths)
* Neutral sixth (-0.5 fifths)
* Neutral seventh (+1.5 fifths)
* Semiaugmented octave (+3.5 fifths)

Intervals are named on a per-octave basis (that is, by octave-reducing, naming the interval, and adding back octaves according to conventional interval arithmetic), so the semidiminished unison and semiaugmented octave (which are lesser than and greater than the unison and octave respectively) do not actually appear in any interval names. Instead, they are chosen to ensure that the boundary between "unison" and "second" always falls precisely halfway between the perfect unison and the minor second.

These intervals may not exist in an edo (for instance, if it maps the fifth to an odd number of steps). This is okay, as they are being used as points of reference to compare to, not as actual necessary steps in the edo.

== Interval regions ==
Each neutral interval defines a series of regions (or "qualities") extending outwards from it, which are defined in terms of equal divisions of [[15/14]]. The use of 15/14 was proposed by [[User:Lériendil|Lériendil]] for threefold reasons:
* Firstly, 15/14 is a mapping of the [[apotome]] in [[aberschismic]] tunings: that is, it is the interval between [[7/6]] and [[5/4]] and between [[6/5]] and [[9/7]], and therefore the interval between the midpoint of 7/6 and 6/5, and the midpoint of 6/5 and 9/7;
* Secondly, it is close to 120 cents, which is the maximum amount of separation an interval can have from a diatonic neutral (assuming the fifth does indeed generate a diatonic scale), ensuring all intervals can be named;
* Finally, it is not itself an equal division of the octave, ensuring that no EDO intervals (aside from the true neutrals) land on region boundaries.
{| class="wikitable"
|+
!\25ed(15/14)
!Cents
!Major
!Minor
|-
|0
|0
| colspan="2" |neutral
|-
|0-2
|0-9.6
|tendoneutral
|artoneutral
|-
|2-5
|9.6-23.9
|submajor
|supraminor
|-
|5-10
|23.9-47.8
|nearmajor
|nearminor
|-
|10-15
|47.8-71.7
|farmajor
|farminor
|-
|15-20
|71.7-95.6
|supermajor
|subminor
|-
|20+
|95.6+
|ultramajor
|inframinor
|}
For instance, assuming a fifth is tuned to JI, the categories of thirds are found at <255c (inframinor), 256-279c (subminor), 280-303c (farminor), 304-327c (nearminor), 327-341c (supraminor), 342-360c (neutral, arto/tendo-), 361-375c (submajor), 376-398c (nearmajor), 399-422c (farmajor), 423-446c (supermajor), and >446c (ultramajor).

With these, the complete sets of intervals of each edo may be given a name. When an interval is an equal distance from two neutrals, thirds are always given precedence over fourths (so that an interval equidistant between the neutral third and neutral fourth is always a kind of third), and over seconds, which take precedence over unisons (except for the perfect unison and octave). The same rules apply to the complementary region of the octave. Fourths always take precedence below the tritone, and fifths always take precedence above it.

The exception is when the diatonic intervals coincide, in which case the conflated interval belongs to the category corresponding to its simplest diatonic interpretation (i.e. 240c is a second, not a third, and 480c is a fourth, not a third or (diminished) fifth). The same applies to oneirotonic and antidiatonic structures.

If there is only one kind of major or minor, drop all prefixes on major and minor. For example, if the only interval qualities found are "farminor", "neutral", and "farmajor", then rename "farminor" to "minor" and "farmajor" to "major". As a result, skip step 3.

== Disambiguation ==
In large edos, multiple intervals may be assigned the same name at the current point. This is where the disambiguation scheme comes into play. Based on the number of intervals in each category, a fixed set of names is assigned in order of size.
{| class="wikitable"
|+
!Quality
!2
!3
|-
|inframinor
|arto, inframinor
|
|-
|subminor
|sensaminor, gothminor
|sensaminor, septiminor, gothminor
|-
|farminor
|neominor, novaminor
|neominor, triminor, novaminor
|-
|nearminor
|valaminor, magiminor
|valaminor, pentaminor, magiminor
|-
|supraminor
|daemominor, aurominor
|
|-
|artoneutral
|subneutral, artoneutral
|
|-
|tendoneutral
|tendoneutral, supraneutral
|
|-
|submajor
|auromajor, daemomajor
|
|-
|nearmajor
|magimajor, valamajor
|magimajor, pentamajor, valamajor
|-
|farmajor
|novamajor, neomajor
|novamajor, trimajor, neomajor
|-
|supermajor
|gothmajor*, sensamajor
|gothmajor*, septimajor, sensamajor
|-
|ultramajor
|ultramajor, tendo
|
|}

In cases where there are two intervals belonging to the nearminor/major, farminor/major, and subminor/supermajor qualities, "pentamajor", "trimajor", and "septimajor" are substituted in for major thirds within 4.78{{c}} (1\25ed(15/14)) of the characteristic just intervals 5/4, 19/15, and 9/7 respectively**. If any major third acquires one of these subqualities, it is then propagated to its complement and other interval degrees.

=== Alternative system ===
Primarily in the case of tuning systems other than EDOs, or large EDOs where more than 3 intervals exist within the space of a single quality band, another fallback system can be used to assign subqualities to specific intervals.

"Trimajor" is defined as a radius of 1\25ed(15/14) around 19/15**, the same way as it is above. "Septimajor" then directly occupies the band 1\5ed(15/14) sharp of trimajor, while "pentamajor" occupies the band 9\50ed(15/14) flat of trimajor. The sharp edge of pentamajor is then taken to be the edge between auromajor and daemomajor. Subneutral and supraneutral intervals are not distinguished in this system.

Pentamajor and septimajor can variantly be defined to center around 5/4 and 9/7 as above, for the sake of consistency with the system generally employed for EDOs.

In cent values, with a justly tuned 3/2, the subqualities sharpward of the neutral third are then bounded as follows:
* 350.978 <- tendoneutral -> 360.533 <- auromajor -> 368.634 <- daemomajor -> 374.866
* 374.866 <- magimajor -> 382.967 <- pentamajor -> 392.522 <- valamajor -> 398.755
* 398.755 <- novamajor -> 404.467 <- trimajor -> 414.022 <- neomajor -> 422.643
* 422.643 <- shrubmajor* -> 428.355 <- septimajor -> 437.911 <- sensamajor -> 446.532

In that case, two intervals falling within the same subquality can then be disambiguated as "small" and "large", or three as "small", "mid", and "large".

* "Shrub-" can be replaced with "goth-".

** A variation would be for 5/4, 19/15, and 9/7 to be substituted here with sqrt(25/24), sqrt(722/675), and sqrt(54/49) above the neutral third, snapping all subqualities to the same positions relative the neutral third.

== Final steps ==
There are some additional replacements to be done:

1) Examine the diatonic fourth and whether it is major or minor. Remove the corresponding quality from all fourth names (for example, if the diatonic fourth is a farminor fourth, replace all instances of "minor fourth" with simply "fourth". Rename the opposing quality from "major" to "augmented", or "minor" to "diminished". If the fourth is any kind of neutral, no change is necessary to any interval names.

2) Label the diatonic fourth "perfect fourth" regardless of its quality.

3) Repeat for the unison, fifth, and octave.

3a) The result may create ambiguities with terms like "far octave" in some edos (the smallest edo to feature this problem being 26edo, between 25\26 and 27\26). In that case, restore "major" to octaves, fifteenths, etc above their perfect counterparts and which have ambiguous labels, and "minor" to fifteenths and above.

4) If quality is not necessary to distinguish intervals at all, remove it entirely (i.e. if there are only neutral intervals, do not specify "neutral").

== Qualities in small diatonic EDOs ==
Below lists the palettes of neutral and major qualities (noting that minor qualities always exist as the complements of major qualities) that can be found in diatonic EDOs below about 60, that is, the EDOs that do not require the disambiguation step. A few EDOs have two diatonic fifths, one which is divisible in two and one which is not. Both fifths are kept track of, but non-patent fifths are in parentheses.

Ultramajor qualities are treated separately, since they are ambiguous in degree. However, for EDOs with flat fifths ([[19edo]] or flatter) and which divide the perfect fourth in two, subminor and supermajor qualities are in fact interordinal (e.g. supermajor thirds are the same as sub(minor) fourths). These EDOs will be marked with an asterisk. Some EDOs with sharp fifths have ultramajor (and inframinor) intervals which are, however, not interordinal; these will be marked with a superscript plus sign.

=== Without a neutral third ===
EDOs without a neutral third have:
* with a step size 21.25-27.3{{c}} -> '''submajor, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[46edo|46]], [[47edo|47]]*, [[49edo|49]], [[50edo|50]], [[53edo|53]], [[56edo|56]]+ ([[52edo|52b]]+, [[54edo|54b]])

* with a step size 27.3-28.65{{c}} -> '''submajor, nearmajor, farmajor'''
** diatonic fifths: [[42edo|42]]+, [[43edo|43]]

* with a step size 28.65-31.85{{c}} -> '''submajor, nearmajor, supermajor'''
** diatonic fifths: [[39edo|39]], [[40edo|40]]

* with a step size 31.85-38.2{{c}} -> '''submajor, farmajor, supermajor'''
** diatonic fifths: [[32edo|32]], [[33edo|33]]*, [[36edo|36]]

* with a step size 38.2-47.8{{c}} -> '''submajor, farmajor'''
** diatonic fifths: [[26edo|26]], [[29edo|29]]

* with a step size 47.8-63.7{{c}} -> '''nearmajor, supermajor'''
** diatonic fifths: [[19edo|19]]*, [[22edo|22]]

* with a step size > 63.7{{c}} -> '''major'''
** diatonic fifths: [[12edo|12]]

=== With a neutral third ===
EDOs with a neutral third have:
* with a step size 19.1-23.9c -> '''neutral, submajor, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[51edo|51]], [[52edo|52]]*, [[54edo|54]], [[55edo|55]], ''[[58edo|58]]'', [[61edo|61]]+, [[62edo|62]] (''[[57edo|57b]]''+, [[59edo|59b]])

* with a step size 23.9-31.85c -> '''neutral, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[38edo|38]]*, [[41edo|41]], [[44edo|44]], [[45edo|45]], [[48edo|48]] ([[47edo|47b]]+)

* with a step size 31.85-35.85c -> '''neutral, nearmajor, farmajor'''
** diatonic fifths: [[34edo|34]], [[37edo|37]]+

* with a step size 35.85-47.8c -> '''neutral, nearmajor, supermajor'''
** diatonic fifths: [[27edo|27]], [[31edo|31]]

* with a step size 47.8-71.65c -> '''neutral, (far)major'''
** diatonic fifths: [[17edo|17]], [[24edo|24]]

== Notes ==
The first EDO this system fails to name the intervals for is currently 159edo, as it has four intervals within each supermajor range.

== Extensions ==

=== Oneirotonic ===
Add an extra ordinal for "tritone" rather than just treating it as a special case for even edos. The chroma is the moschroma of oneirotonic.

=== Antidiatonic ===
The chroma is the moschroma of antidiatonic. Note that the pythagorean semidiminished unison is still the center of the unison range, despite being larger than 0c.

Adaptive diatonic interval names

2026-05-25T19:41:58Z

Lériendil: added links to the edos

The system of '''adaptive diatonic interval names (ADIN)''', developed by Vector and Leriendil, is a way to (mostly) uniquely label the intervals in an EDO based on size and relation to that EDO's patent fifth. It is ''diatonic'' because it attempts to behave predictably relative to MOSdiatonic staff notation, and it is ''adaptive'' because the differing qualities of diatonic intervals in different tunings are reflected in the interval names (that is to say, it "adapts" to different diatonic tunings). Finally, it is an ''interval naming system'', not a notation system, because it provides no way to write notes and labels intervals based on "what they are", not "what they do". (The creator of the ADIN system endorses [[Modified ups and downs notation|ups and downs notation]] for the latter.)

It is an attempt at formalizing the systems of interval qualities used by various xenharmonic resources on the internet.

=== On "major" vs. "supermajor" ===
A large number of resources, including Unque's theory page on xen.wiki, the Lumatone introductory video to 22edo, and the 31et.com page on 22edo, alongside resources for temperaments such as 41edo (31et.com) and 27edo (Lumatone), reserve "major/minor" for 5/4 and 6/5, distinguishing an unmarked "major" from terms like "supermajor" (and "minor" from terms like "subminor"). There is no obvious reason to do this other than 5-limit preferentialism, and it leads to ambiguity where "major" could either refer to specifically classical major intervals or more generally to any major interval. This is especially egregious as in [[Archy|archytas]] temperaments (a major form of structural temperament for the 7-limit), diatonic notation has the opposite behavior, leaving 9/7 and 7/6 unmarked while 5/4 and 6/5 get the extra prefix, meanwhile standard diatonic notation in just intonation, schismic, hemifamity, etc. uses "major" to refer to neither, instead denoting an interval in between the two, leaving only meantone temperaments unambiguous.

While the use of "major" in standard diatonic notation is not a problem on its own, it leads to a large degree of ambiguity with naming schemes wherein 5/4 is prioritized outside of meantone temperaments. To resolve this, ADIN provides the original label "nearmajor" for intervals with a similar quality to 5/4, and "nearminor" for intervals with a similar quality to 6/5. Additionally, "farmajor" and "farminor" are used to refer to intervals in the standard JI diatonic range, regardless of the actual tuning of the diatonic scale. Nearminor and nearmajor intervals may otherwise be called "classic(al)", "pental", or "ptolemaic" minor/major, which are terms used to describe the simple 5-limit intervals to which they correspond.

Unqualified "major" may refer to the range of major qualities collectively in cases like "either major key" or "the major thirds". However, when used to refer to a specific interval ("the major third"), it should refer to specifically the MOS diatonic intervals. Likewise for minor.

== Premise ==
ADIN names qualities, and then applies those names to intervals based on their distance from the nearest (possibly imaginary) diatonic neutral interval. The diatonic neutral intervals are as follows:

* Semidiminished unison (-3.5 fifths)
* Neutral second (-1.5 fifths)
* Neutral third (+0.5 fifths)
* Semiaugmented fourth (+2.5 fifths)
* Semidiminished fifth (-2.5 fifths)
* Neutral sixth (-0.5 fifths)
* Neutral seventh (+1.5 fifths)
* Semiaugmented octave (+3.5 fifths)

Intervals are named on a per-octave basis (that is, by octave-reducing, naming the interval, and adding back octaves according to conventional interval arithmetic), so the semidiminished unison and semiaugmented octave (which are lesser than and greater than the unison and octave respectively) do not actually appear in any interval names. Instead, they are chosen to ensure that the boundary between "unison" and "second" always falls precisely halfway between the perfect unison and the minor second.

These intervals may not exist in an edo (for instance, if it maps the fifth to an odd number of steps). This is okay, as they are being used as points of reference to compare to, not as actual necessary steps in the edo.

== Interval regions ==
Each neutral interval defines a series of regions (or "qualities") extending outwards from it, which are defined in terms of equal divisions of [[15/14]]. The use of 15/14 was proposed by [[User:Lériendil|Lériendil]] for threefold reasons:
* Firstly, 15/14 is a mapping of the [[apotome]] in [[aberschismic]] tunings: that is, it is the interval between [[7/6]] and [[5/4]] and between [[6/5]] and [[9/7]], and therefore the interval between the midpoint of 7/6 and 6/5, and the midpoint of 6/5 and 9/7;
* Secondly, it is close to 120 cents, which is the maximum amount of separation an interval can have from a diatonic neutral (assuming the fifth does indeed generate a diatonic scale), ensuring all intervals can be named;
* Finally, it is not itself an equal division of the octave, ensuring that no EDO intervals (aside from the true neutrals) land on region boundaries.
{| class="wikitable"
|+
!\25ed(15/14)
!Cents
!Major
!Minor
|-
|0
|0
| colspan="2" |neutral
|-
|0-2
|0-9.6
|tendoneutral
|artoneutral
|-
|2-5
|9.6-23.9
|submajor
|supraminor
|-
|5-10
|23.9-47.8
|nearmajor
|nearminor
|-
|10-15
|47.8-71.7
|farmajor
|farminor
|-
|15-20
|71.7-95.6
|supermajor
|subminor
|-
|20+
|95.6+
|ultramajor
|inframinor
|}
For instance, assuming a fifth is tuned to JI, the categories of thirds are found at <255c (inframinor), 256-279c (subminor), 280-303c (farminor), 304-327c (nearminor), 327-341c (supraminor), 342-360c (neutral, arto/tendo-), 361-375c (submajor), 376-398c (nearmajor), 399-422c (farmajor), 423-446c (supermajor), and >446c (ultramajor).

With these, the complete sets of intervals of each edo may be given a name. When an interval is an equal distance from two neutrals, thirds are always given precedence over fourths (so that an interval equidistant between the neutral third and neutral fourth is always a kind of third), and over seconds, which take precedence over unisons (except for the perfect unison and octave). The same rules apply to the complementary region of the octave. Fourths always take precedence below the tritone, and fifths always take precedence above it.

The exception is when the diatonic intervals coincide, in which case the conflated interval belongs to the category corresponding to its simplest diatonic interpretation (i.e. 240c is a second, not a third, and 480c is a fourth, not a third or (diminished) fifth). The same applies to oneirotonic and antidiatonic structures.

If there is only one kind of major or minor, drop all prefixes on major and minor. For example, if the only interval qualities found are "farminor", "neutral", and "farmajor", then rename "farminor" to "minor" and "farmajor" to "major". As a result, skip step 3.

== Disambiguation ==
In large edos, multiple intervals may be assigned the same name at the current point. This is where the disambiguation scheme comes into play. Based on the number of intervals in each category, a fixed set of names is assigned in order of size.
{| class="wikitable"
|+
!Quality
!2
!3
|-
|inframinor
|arto, inframinor
|
|-
|subminor
|sensaminor, gothminor
|sensaminor, septiminor, gothminor
|-
|farminor
|neominor, novaminor
|neominor, triminor, novaminor
|-
|nearminor
|valaminor, magiminor
|valaminor, pentaminor, magiminor
|-
|supraminor
|daemominor, aurominor
|
|-
|artoneutral
|subneutral, artoneutral
|
|-
|tendoneutral
|tendoneutral, supraneutral
|
|-
|submajor
|auromajor, daemomajor
|
|-
|nearmajor
|magimajor, valamajor
|magimajor, pentamajor, valamajor
|-
|farmajor
|novamajor, neomajor
|novamajor, trimajor, neomajor
|-
|supermajor
|gothmajor*, sensamajor
|gothmajor*, septimajor, sensamajor
|-
|ultramajor
|ultramajor, tendo
|
|}

In cases where there are two intervals belonging to the nearminor/major, farminor/major, and subminor/supermajor qualities, "pentamajor", "trimajor", and "septimajor" are substituted in for major thirds within 4.78{{c}} (1\25ed(15/14)) of the characteristic just intervals 5/4, 19/15, and 9/7 respectively**. If any major third acquires one of these subqualities, it is then propagated to its complement and other interval degrees.

=== Alternative system ===
Primarily in the case of tuning systems other than EDOs, or large EDOs where more than 3 intervals exist within the space of a single quality band, another fallback system can be used to assign subqualities to specific intervals.

"Trimajor" is defined as a radius of 1\25ed(15/14) around 19/15**, the same way as it is above. "Septimajor" then directly occupies the band 1\5ed(15/14) sharp of trimajor, while "pentamajor" occupies the band 9\50ed(15/14) flat of trimajor. The sharp edge of pentamajor is then taken to be the edge between auromajor and daemomajor. Subneutral and supraneutral intervals are not distinguished in this system.

Pentamajor and septimajor can variantly be defined to center around 5/4 and 9/7 as above, for the sake of consistency with the system generally employed for EDOs.

In cent values, with a justly tuned 3/2, the subqualities sharpward of the neutral third are then bounded as follows:
* 350.978 <- tendoneutral -> 360.533 <- auromajor -> 368.634 <- daemomajor -> 374.866
* 374.866 <- magimajor -> 382.967 <- pentamajor -> 392.522 <- valamajor -> 398.755
* 398.755 <- novamajor -> 404.467 <- trimajor -> 414.022 <- neomajor -> 422.643
* 422.643 <- shrubmajor* -> 428.355 <- septimajor -> 437.911 <- sensamajor -> 446.532

In that case, two intervals falling within the same subquality can then be disambiguated as "small" and "large", or three as "small", "mid", and "large".

* "Shrub-" can be replaced with "goth-".

** A variation would be for 5/4, 19/15, and 9/7 to be substituted here with sqrt(25/24), sqrt(722/675), and sqrt(54/49) above the neutral third, snapping all subqualities to the same positions relative the neutral third.

== Final steps ==
There are some additional replacements to be done:

1) Examine the diatonic fourth and whether it is major or minor. Remove the corresponding quality from all fourth names (for example, if the diatonic fourth is a farminor fourth, replace all instances of "minor fourth" with simply "fourth". Rename the opposing quality from "major" to "augmented", or "minor" to "diminished". If the fourth is any kind of neutral, no change is necessary to any interval names.

2) Label the diatonic fourth "perfect fourth" regardless of its quality.

3) Repeat for the unison, fifth, and octave.

3a) The result may create ambiguities with terms like "far octave" in some edos (the smallest edo to feature this problem being 26edo, between 25\26 and 27\26). In that case, restore "major" to octaves, fifteenths, etc above their perfect counterparts and which have ambiguous labels, and "minor" to fifteenths and above.

4) If quality is not necessary to distinguish intervals at all, remove it entirely (i.e. if there are only neutral intervals, do not specify "neutral").

== Qualities in small diatonic EDOs ==
Below lists the palettes of neutral and major qualities (noting that minor qualities always exist as the complements of major qualities) that can be found in diatonic EDOs below about 60, that is, the EDOs that do not require the disambiguation step. A few EDOs have two diatonic fifths, one which is divisible in two and one which is not. Both fifths are kept track of, but non-patent fifths are in parentheses.

Ultramajor qualities are treated separately, since they are ambiguous in degree. However, for EDOs with flat fifths ([[19edo]] or flatter) and which divide the perfect fourth in two, subminor and supermajor qualities are in fact interordinal (e.g. supermajor thirds are the same as sub(minor) fourths). These EDOs will be marked with an asterisk. Some EDOs with sharp fifths have ultramajor (and inframinor) intervals which are, however, not interordinal; these will be marked with a superscript plus sign.

=== Without a neutral third ===
EDOs without a neutral third have:
* with a step size 21.25-27.3{{c}} -> '''submajor, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[46edo|46]], [[47edo|47]]*, [[49edo|49]], [[50edo|50]], [[53edo|53]], [[56edo|56]]+ ([[52edo|52b]]+, [[54edo|54b]])

* with a step size 27.3-28.65{{c}} -> '''submajor, nearmajor, farmajor'''
** diatonic fifths: [[42edo|42]]+, [[43edo|43]]

* with a step size 28.65-31.85{{c}} -> '''submajor, nearmajor, supermajor'''
** diatonic fifths: [[39edo|39]], [[40edo|40]]

* with a step size 31.85-38.2{{c}} -> '''submajor, farmajor, supermajor'''
** diatonic fifths: [[32edo|32]], [[33edo|33]]*, [[36edo|36]]

* with a step size 38.2-47.8{{c}} -> '''submajor, farmajor'''
** diatonic fifths: [[26edo|26]], [[29edo|29]]

* with a step size 47.8-63.7{{c}} -> '''nearmajor, supermajor'''
** diatonic fifths: [[19edo|19]]*, [[22edo|22]]

* with a step size > 63.7{{c}} -> '''major'''
** diatonic fifths: [[12edo|12]]

=== With a neutral third ===
EDOs with a neutral third have:
* with a step size 19.1-23.9c -> '''neutral, submajor, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[51edo|51]], [[52edo|52]]*, [[54edo|54]], [[55edo|55]], [[58edo|58]], [[61edo|61]]+, [[62edo|62]] ([[57edo|57b]]+, [[59edo|59b]])

* with a step size 23.9-31.85c -> '''neutral, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[38edo|38]]*, [[41edo|41]], [[44edo|44]], [[45edo|45]], [[48edo|48]] ([[47edo|47b]]+)

* with a step size 31.85-35.85c -> '''neutral, nearmajor, farmajor'''
** diatonic fifths: [[34edo|34]], [[37edo|37]]+

* with a step size 35.85-47.8c -> '''neutral, nearmajor, supermajor'''
** diatonic fifths: [[27edo|27]], [[31edo|31]]

* with a step size 47.8-71.65c -> '''neutral, (far)major'''
** diatonic fifths: [[17edo|17]], [[24edo|24]]

== Notes ==
The first EDO this system fails to name the intervals for is currently 159edo, as it has four intervals within each supermajor range.

== Extensions ==

=== Oneirotonic ===
Add an extra ordinal for "tritone" rather than just treating it as a special case for even edos. The chroma is the moschroma of oneirotonic.

=== Antidiatonic ===
The chroma is the moschroma of antidiatonic. Note that the pythagorean semidiminished unison is still the center of the unison range, despite being larger than 0c.

User:Lériendil/common.css

2026-05-24T15:29:35Z

Lériendil: reverted to the Old Lériendil color scheme for light mode

/* class="adv" selector, TODO finish formatting */
.adv {
opacity: 0.60;
}

/* Style in Visual Editor */
.oo-ui-labelElement .oo-ui-labelElement-label {
color: #888 !important;
}
.oo-ui-tool-title {
color: #000 !important;
}
.oo-ui-toolbar-bar {
color: #fff !important;
}
.ve-ui-symbolListPage h3 {
color: #fff !important;
}
.ve-ui-mwLatexDialog-symbol {
color: #fff !important;
}
.ve-ui-symbolListPage-symbol {
color: #fff !important;
}

.mw-content-ltr.mw-highlight-lines pre, .mw-content-ltr.content .mw-highlight-lines pre {
box-shadow: inset 2.75em 0 0 #000;
}

.mw-highlight {
background: #000;
}

@media (prefers-color-scheme: light) {
.thl, .wikitable .thl {
color: #000 !important;
background-color: #cba !important;
}
.prime2 {
background-color: #BFBFBF !important;
color: black !important;
}
.prime3 {
background-color: #EE5F64 !important;
color: black !important;
}
.prime5 {
background-color: #C5FE95 !important;
color: black !important;
}
.prime7 {
background-color: #7F5FC5 !important;
color: black !important;
}
.prime11 {
background-color: #FFDD71 !important;
color: black !important;
}
.prime13 {
background-color: #D35FD5 !important;
color: black !important;
}
.prime17 {
background-color: #79D8FE !important;
color: black !important;
}
.prime19 {
background-color: #98FEB7 !important;
color: black !important;
}
.prime23 {
background-color: #FCAF60 !important;
color: black !important;
}
.prime29 {
background-color: #5F69C5 !important;
color: black !important;
}
.prime31 {
background-color: #5F99D3 !important;
color: black !important;
}
.prime37 {
background-color: #9DEAD8 !important;
color: black !important;
}
.prime41 {
background-color: #D6EA95 !important;
color: black !important;
}
.prime43 {
background-color: #EADB8C !important;
color: black !important;
}
.prime47 {
background-color: #E99372 !important;
color: black !important;
}
.prime53 {
background-color: #B272C9 !important;
color: black !important;
}
.acc0 {
background-color: #EBF !important;
color: black !important;
}
.acc1 {
background-color: #BBEEEE !important;
color: black !important;
}
.acc2 {
background-color: #BBFFD3 !important;
color: black !important;
}
.acc3 {
background-color: #CCFFBB !important;
color: black !important;
}
.acc4 {
background-color: #E8FFBB !important;
color: black !important;
}
.acc5 {
background-color: #FFF8BB !important;
color: black !important;
}
.acc6 {
background-color: #FFD8BB !important;
color: black !important;
}
.acc7 {
background-color: #FFBBBB !important;
color: black !important;
}
}

@media (prefers-color-scheme: dark) {
.thl, .wikitable .thl {
color: #fff !important;
background-color: #531 !important;
}
.prime2 {
background-color: #3F3F3F !important;
color: white !important;
}
.prime3 {
background-color: #8F0005 !important;
color: white !important;
}
.prime5 {
background-color: #669F36 !important;
color: white !important;
}
.prime7 {
background-color: #200066 !important;
color: white !important;
}
.prime11 {
background-color: #A07E12 !important;
color: white !important;
}
.prime13 {
background-color: #740076 !important;
color: white !important;
}
.prime17 {
background-color: #247090 !important;
color: white !important;
}
.prime19 {
background-color: #3C9056 !important;
color: white !important;
}
.prime23 {
background-color: #8F4F0F !important;
color: white !important;
}
.prime29 {
background-color: #0E1662 !important;
color: white !important;
}
.prime31 {
background-color: #0E3E6D !important;
color: white !important;
}
.prime37 {
background-color: #418172 !important;
color: white !important;
}
.prime41 {
background-color: #70813A !important;
color: white !important;
}
.prime43 {
background-color: #817532 !important;
color: white !important;
}
.prime47 {
background-color: #81391D !important;
color: white !important;
}
.prime53 {
background-color: #521D66 !important;
color: white !important;
}
.acc0 {
background-color: #304 !important;
color: white !important;
}
.acc1 {
background-color: #033 !important;
color: white !important;
}
.acc2 {
background-color: #004418 !important;
color: white !important;
}
.acc3 {
background-color: #140 !important;
color: white !important;
}
.acc4 {
background-color: #2D4400 !important;
color: white !important;
}
.acc5 {
background-color: #443D00 !important;
color: white !important;
}
.acc6 {
background-color: #441D00 !important;
color: white !important;
}
.acc7 {
background-color: #440000 !important;
color: white !important;
}
}

MediaWiki:Common.css

2026-05-24T15:28:27Z

Lériendil: rolled out new color schemes for both bright and dark

/* class="adv" selector, TODO finish formatting */
.adv {
opacity: 0.60;
}

/* Style in Visual Editor */
.oo-ui-labelElement .oo-ui-labelElement-label {
color: #888 !important;
}
.oo-ui-tool-title {
color: #000 !important;
}
.oo-ui-toolbar-bar {
color: #fff !important;
}
.ve-ui-symbolListPage h3 {
color: #fff !important;
}
.ve-ui-mwLatexDialog-symbol {
color: #fff !important;
}
.ve-ui-symbolListPage-symbol {
color: #fff !important;
}

.mw-content-ltr.mw-highlight-lines pre, .mw-content-ltr.content .mw-highlight-lines pre {
box-shadow: inset 2.75em 0 0 #000;
}

.mw-highlight {
background: #000;
}

@media (prefers-color-scheme: light) {
.thl, .wikitable .thl {
color: #000 !important;
background-color: #cba !important;
}
.prime2 {
background-color: #BFBFBF !important;
color: black !important;
}
.prime3 {
background-color: #EE5F64 !important;
color: black !important;
}
.prime5 {
background-color: #C5FE95 !important;
color: black !important;
}
.prime7 {
background-color: #7F5FC5 !important;
color: black !important;
}
.prime11 {
background-color: #FFDD71 !important;
color: black !important;
}
.prime13 {
background-color: #D35FD5 !important;
color: black !important;
}
.prime17 {
background-color: #83CFEF !important;
color: black !important;
}
.prime19 {
background-color: #9BEFB5 !important;
color: black !important;
}
.prime23 {
background-color: #EEAE6E !important;
color: black !important;
}
.prime29 {
background-color: #6D75C1 !important;
color: black !important;
}
.prime31 {
background-color: #6D9DCC !important;
color: black !important;
}
.prime37 {
background-color: #A0E0D1 !important;
color: black !important;
}
.prime41 {
background-color: #CFE099 !important;
color: black !important;
}
.prime43 {
background-color: #E0D491 !important;
color: black !important;
}
.prime47 {
background-color: #E0987C !important;
color: black !important;
}
.prime53 {
background-color: #B17CC5 !important;
color: black !important;
}
.acc0 {
background-color: #EBF !important;
color: black !important;
}
.acc1 {
background-color: #BBEEEE !important;
color: black !important;
}
.acc2 {
background-color: #BBFFD3 !important;
color: black !important;
}
.acc3 {
background-color: #CCFFBB !important;
color: black !important;
}
.acc4 {
background-color: #E8FFBB !important;
color: black !important;
}
.acc5 {
background-color: #FFF8BB !important;
color: black !important;
}
.acc6 {
background-color: #FFD8BB !important;
color: black !important;
}
.acc7 {
background-color: #FFBBBB !important;
color: black !important;
}
}

@media (prefers-color-scheme: dark) {
.thl, .wikitable .thl {
color: #fff !important;
background-color: #531 !important;
}
.prime2 {
background-color: #3F3F3F !important;
color: white !important;
}
.prime3 {
background-color: #8F0005 !important;
color: white !important;
}
.prime5 {
background-color: #669F36 !important;
color: white !important;
}
.prime7 {
background-color: #200066 !important;
color: white !important;
}
.prime11 {
background-color: #A07E12 !important;
color: white !important;
}
.prime13 {
background-color: #740076 !important;
color: white !important;
}
.prime17 {
background-color: #247090 !important;
color: white !important;
}
.prime19 {
background-color: #3C9056 !important;
color: white !important;
}
.prime23 {
background-color: #8F4F0F !important;
color: white !important;
}
.prime29 {
background-color: #0E1662 !important;
color: white !important;
}
.prime31 {
background-color: #0E3E6D !important;
color: white !important;
}
.prime37 {
background-color: #418172 !important;
color: white !important;
}
.prime41 {
background-color: #70813A !important;
color: white !important;
}
.prime43 {
background-color: #817532 !important;
color: white !important;
}
.prime47 {
background-color: #81391D !important;
color: white !important;
}
.prime53 {
background-color: #521D66 !important;
color: white !important;
}
.acc0 {
background-color: #304 !important;
color: white !important;
}
.acc1 {
background-color: #033 !important;
color: white !important;
}
.acc2 {
background-color: #004418 !important;
color: white !important;
}
.acc3 {
background-color: #140 !important;
color: white !important;
}
.acc4 {
background-color: #2D4400 !important;
color: white !important;
}
.acc5 {
background-color: #443D00 !important;
color: white !important;
}
.acc6 {
background-color: #441D00 !important;
color: white !important;
}
.acc7 {
background-color: #440000 !important;
color: white !important;
}
}

User:Lériendil/common.css

2026-05-24T15:27:53Z

Lériendil: rolled out new color schemes for both bright and dark

/* class="adv" selector, TODO finish formatting */
.adv {
opacity: 0.60;
}

/* Style in Visual Editor */
.oo-ui-labelElement .oo-ui-labelElement-label {
color: #888 !important;
}
.oo-ui-tool-title {
color: #000 !important;
}
.oo-ui-toolbar-bar {
color: #fff !important;
}
.ve-ui-symbolListPage h3 {
color: #fff !important;
}
.ve-ui-mwLatexDialog-symbol {
color: #fff !important;
}
.ve-ui-symbolListPage-symbol {
color: #fff !important;
}

.mw-content-ltr.mw-highlight-lines pre, .mw-content-ltr.content .mw-highlight-lines pre {
box-shadow: inset 2.75em 0 0 #000;
}

.mw-highlight {
background: #000;
}

@media (prefers-color-scheme: light) {
.thl, .wikitable .thl {
color: #000 !important;
background-color: #cba !important;
}
.prime2 {
background-color: #BFBFBF !important;
color: black !important;
}
.prime3 {
background-color: #EE5F64 !important;
color: black !important;
}
.prime5 {
background-color: #C5FE95 !important;
color: black !important;
}
.prime7 {
background-color: #7F5FC5 !important;
color: black !important;
}
.prime11 {
background-color: #FFDD71 !important;
color: black !important;
}
.prime13 {
background-color: #D35FD5 !important;
color: black !important;
}
.prime17 {
background-color: #83CFEF !important;
color: black !important;
}
.prime19 {
background-color: #9BEFB5 !important;
color: black !important;
}
.prime23 {
background-color: #EEAE6E !important;
color: black !important;
}
.prime29 {
background-color: #6D75C1 !important;
color: black !important;
}
.prime31 {
background-color: #6D9DCC !important;
color: black !important;
}
.prime37 {
background-color: #A0E0D1 !important;
color: black !important;
}
.prime41 {
background-color: #CFE099 !important;
color: black !important;
}
.prime43 {
background-color: #E0D491 !important;
color: black !important;
}
.prime47 {
background-color: #E0987C !important;
color: black !important;
}
.prime53 {
background-color: #B17CC5 !important;
color: black !important;
}
.acc0 {
background-color: #EBF !important;
color: black !important;
}
.acc1 {
background-color: #BBEEEE !important;
color: black !important;
}
.acc2 {
background-color: #BBFFD3 !important;
color: black !important;
}
.acc3 {
background-color: #CCFFBB !important;
color: black !important;
}
.acc4 {
background-color: #E8FFBB !important;
color: black !important;
}
.acc5 {
background-color: #FFF8BB !important;
color: black !important;
}
.acc6 {
background-color: #FFD8BB !important;
color: black !important;
}
.acc7 {
background-color: #FFBBBB !important;
color: black !important;
}
}

@media (prefers-color-scheme: dark) {
.thl, .wikitable .thl {
color: #fff !important;
background-color: #531 !important;
}
.prime2 {
background-color: #3F3F3F !important;
color: white !important;
}
.prime3 {
background-color: #8F0005 !important;
color: white !important;
}
.prime5 {
background-color: #669F36 !important;
color: white !important;
}
.prime7 {
background-color: #200066 !important;
color: white !important;
}
.prime11 {
background-color: #A07E12 !important;
color: white !important;
}
.prime13 {
background-color: #740076 !important;
color: white !important;
}
.prime17 {
background-color: #247090 !important;
color: white !important;
}
.prime19 {
background-color: #3C9056 !important;
color: white !important;
}
.prime23 {
background-color: #8F4F0F !important;
color: white !important;
}
.prime29 {
background-color: #0E1662 !important;
color: white !important;
}
.prime31 {
background-color: #0E3E6D !important;
color: white !important;
}
.prime37 {
background-color: #418172 !important;
color: white !important;
}
.prime41 {
background-color: #70813A !important;
color: white !important;
}
.prime43 {
background-color: #817532 !important;
color: white !important;
}
.prime47 {
background-color: #81391D !important;
color: white !important;
}
.prime53 {
background-color: #521D66 !important;
color: white !important;
}
.acc0 {
background-color: #304 !important;
color: white !important;
}
.acc1 {
background-color: #033 !important;
color: white !important;
}
.acc2 {
background-color: #004418 !important;
color: white !important;
}
.acc3 {
background-color: #140 !important;
color: white !important;
}
.acc4 {
background-color: #2D4400 !important;
color: white !important;
}
.acc5 {
background-color: #443D00 !important;
color: white !important;
}
.acc6 {
background-color: #441D00 !important;
color: white !important;
}
.acc7 {
background-color: #440000 !important;
color: white !important;
}
}

7-limit

2026-05-22T22:48:45Z

Lériendil: added hemithirds[31]

The '''7-limit''' or the '''2.3.5.7 subgroup''' is the subgroup of [[just intonation]] consisting of the intervals reachable by stacking [[2/1]], [[3/2]], [[5/4]], and [[7/4]]. Important subsets of the 7-limit include the [[7-odd-limit]] and [[9-odd-limit]].

Rank-3 subgroups:
* [[5-limit]]
* [[2.3.7 subgroup]]
* [[2.5.7 subgroup]]
* [[3.5.7 subgroup]]

{{Adv|The 7-limit includes the following odd harmonics below 256: 1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, 105, 125, 135, 147, 175, 189, 225, 243, 245.}}
== Temperaments ==
Some important rank-2 full 7-limit temperaments:
* [[Superpyth]]
* Septimal [[Porcupine]]
* [[Garibaldi]]
* Septimal [[Meantone]]
* [[Meantone#Extensions|Flattone]]
* [[Pajara]]
* Septimal [[Magic]]
* [[Mothra]]
* [[Rodan]]
* [[Valentine]]
* [[Orwell]]
* [[Augene]]
* [[Blackwood]]
* [[Sensi]]

The most important rank-3 full 7-limit temperaments are
* [[Aberschismic]] ({{e|41}} & {{e|46}} & {{e|53}}): equates the 2.3.5 and 2.3.7 formal commas, 81/80 and 64/63; medium-high accuracy.
* [[Marvel]] ({{e|19}} & {{e|22}} & {{e|41}}): equates 25/16 and 14/9; medium accuracy.

== 7-limit interval qualities ==
The 7-limit thirds, in order of stability / consonance in triads (from most to least consonant), are 5/4, 7/6, 6/5, 9/7. Note that 7-limit L/MCJI offers four distinct interval qualities, whereas 12edo and simpler tuning systems tend to offer only two. The mapping of 12edo's interval qualities to the 7-limit's is best understood by considering each 12edo interval as "splitting" into two distinct 7-limit intervals, rather than the alternative approach of retaining 12edo's interval qualities and providing an additional neutral interval (characteristic of subgroups involving 11, like 2.3.11.19).

The four qualities of the 7-limit can be broken down into stable/unstable and bright/dark. The scheme used is the application of ADIN to systems like 22edo and 27edo; see [[Adaptive diatonic interval names#On "major" vs. "supermajor"]] for a further explanation.
{| class="wikitable"
!
!Stable
!Unstable
|-
!Bright
|Nearmajor (warm, pleasant, comforting)
|Supermajor (excited, animated, active)
|-
!Dark
|Subminor (depressive, sad, bluesy)
|Nearminor (angry, tense, stressful)
|}
These are, in fact, the basic "color qualities" provided by Kite: red (ru) = supermajor, yellow (yo) = nearmajor, green (gu) = nearminor, and blue (zo) = subminor. In [[keemic]] temperaments, they become equally spaced, incentivizing the metaphor of a "rainbow" of qualities promoted by Kite.

== Full 7-limit JI scales ==
The scales are shown in [https://sw3.lumipakkanen.com/ Scale Workshop 3] format. Copy and paste into Scale Workshop 3 and you will be able to play the scale.
=== Mode 5 ===
<pre>
8:9:10:12:14:16
</pre>
The simplest full 7-limit JI scale.

=== /2 Mixolydian ===
<pre>
16:18:20:21:24:27:28:32
</pre>

=== /2 Ionian ===
<pre>
16:18:20:21:24:27:30:32
</pre>
=== Zil ===
Zil (from the temperament Godzilla which the zil series serves as a detempering of) is a series of 7-limit JI scales created from a [[generator sequence]] GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160).
==== Zil[14] ====
The most discussed of the zil scales is zil[14] which is chiral depending on the chirality of the [[interleaving|interleaved]] 5-limit [[zarlino]] copies:

RH zil[14] = [[cross-set]] of RH zarlino and 7/4
<pre>
35/32; 9/8; 315/256; 5/4; 21/16; 45/32; 189/128; 3/2; 105/64; 27/16; 7/4; 15/8; 63/32; 2/1
</pre>

LH zil[14] = cross-set of LH zarlino and 7/4
<pre>
21/20; 9/8; 7/6; 6/5; 21/16; 4/3; 7/5; 3/2; 63/40; 8/5; 7/4; 9/5; 63/32; 2/1
</pre>

==== Zil[24] ====
Zil[24] is achiral. It has a 4×3×2 structure in the 7-limit lattice.
<pre>
525/512; 135/128; 35/32; 9/8; 4725/4096; 75/64; 315/256; 5/4; 21/16; 675/512; 175/128; 45/32; 189/128; 3/2; 1575/1024; 25/16; 105/64; 27/16; 7/4; 225/128; 945/512; 15/8; 63/32; 2/1
</pre>

=== Cross-set of 12:14:16:18:21:24 and 5/4 ===
A 10-note scale with an analogous structure to zil[14] (note that these are subsets of both zil[14] chiralities):

RH
<pre>
35/32; 9/8; 5/4; 21/16; 45/32; 3/2; 105/64; 7/4; 15/8; 2/1
</pre>

LH
<pre>
16/15; 8/7; 6/5; 4/3; 48/35; 3/2; 8/5; 12/7; 64/35; 2/1
</pre>

=== A Mothra[36] detemper ===
GS(8/7 8/7 147/128 8/7 8/7 147/128 8/7 8/7 147/128 8/7 8/7 245/216)[36]; 4×3×3 generator structure

<pre>
33075/32768; 525/512; 135/128; 2205/2048; 35/32; 9/8; 147/128; 4725/4096; 75/64; 1225/1024; 315/256; 5/4; 1323/1024; 21/16; 675/512; 11025/8192; 175/128; 45/32; 735/512; 189/128; 3/2; 49/32; 1575/1024; 25/16; 6615/4096; 105/64; 27/16; 441/256; 7/4; 225/128; 3675/2048; 945/512; 15/8; 245/128; 63/32; 2/1
</pre>

=== A Hemithirds[31] detemper ===
<pre>
525/512; 21/20; 16/15; 35/32; 28/25; 8/7; 75/64; 25/21; 128/105; 5/4; 32/25; 21/16; 4/3; 175/128; 7/5; 10/7; 256/175; 3/2; 32/21; 25/16; 8/5; 105/64; 42/25; 128/75; 7/4; 25/14; 64/35; 15/8; 40/21; 1024/525; 2/1
</pre>

=== An Aberschismic 34edo detemper ===
Contains multiple copies of [[aberrisma|aberrismic]] scales (diasem and blackdye); maps both 81/80 and 64/63 to one step.
<pre>
81/80; 21/20; 16/15; 35/32; 10/9; 9/8; 7/6; 189/160; 6/5; 315/256; 5/4; 35/27; 21/16; 4/3; 27/20; 7/5; 45/32; 35/24; 189/128; 3/2; 14/9; 63/40; 8/5; 105/64; 5/3; 27/16; 7/4; 16/9; 9/5; 28/15; 15/8; 35/18; 63/32; 2/1
</pre>

Use 81/64 instead of 35/27 to get a Pyth[7] subset:
<pre>
81/80; 21/20; 16/15; 35/32; 10/9; 9/8; 7/6; 189/160; 6/5; 315/256; 5/4; 81/64; 21/16; 4/3; 27/20; 7/5; 45/32; 35/24; 189/128; 3/2; 14/9; 63/40; 8/5; 105/64; 5/3; 27/16; 7/4; 16/9; 9/5; 28/15; 15/8; 35/18; 63/32; 2/1
</pre>

== Full 7-limit tempered scales ==
=== Superpyth[12] ===
Superpyth[12] is constructed by applying [[Superpyth]] temperament (2.3.5.7[22 & 27]; equivalently tempering out 64/63 and 245/243) to a 12-note chain of fifths. It contains Superpyth-tempered 5-limit [[blackdye]].
<pre>
let L = 2187/2048
let s = 256/243
L;s;L;s;L;s;s;L;s;L;s;s;
stack()
27@
</pre>

=== Pajara ===
[[Pajara]] can be used as an interpretation of 2L8s and 10L2s or their modifications. Pajara works best in [[22edo]].
==== Pajara[10] ====
<pre>
let L = 10/9
let s = 16/15
s;s;L;s;s;s;s;L;s;s;
stack()
22@
</pre>
==== Pentachordal Pajara[10] ====
<pre>
let L = 10/9
let s = 16/15
s;s;s;s;s;L;s;s;s;L;
stack()
22@
</pre>
==== Pajara[12] ====
<pre>
let L = 16/15
let s = 25/24
L;L;L;L;L;s;L;L;L;L;L;s;
stack()
22@
</pre>
==== Hexachordal Pajara[12] ====
<pre>
let L = 16/15
let s = 25/24
L;L;L;L;s;L;L;L;L;L;L;s;
stack()
22@
</pre>
=== 7-limit diachrome ===
7-limit diachrome, an [[aberrismic]] scale, is constructed by taking a 6+6 (for achiral diachrome) or 7+5 (for chiral diachrome) fifth chain structure and tempering out [[5120/5103]]. The scales are shown below in 41edo tuning, but they work in any Aberschismic tuning such as [[46edo]] and [[53edo]].
==== 5sC ====
<pre>
let L = 10/9
let m = 256/243
let s = 81/80
L;s;L;s;L;m;s;L;s;L;s;m;
stack()
41@
</pre>
==== 5sL ====
<pre>
let L = 10/9
let m = 256/243
let s = 81/80
L;s;L;s;L;m;L;s;L;s;m;
stack()
41@
</pre>
==== 5sR ====
<pre>
let L = 10/9
let m = 256/243
let s = 81/80
L;m;s;L;s;L;s;L;m;s;L;s;
stack()
41@
</pre>

=== Aberschismic whitedye ===
Aberschismic whitedye is constructed by taking a diatonic scale and offsetting it by 64/63~81/80, tempering out [[5120/5103]].

Shown below in 41edo tuning.
<pre>
let L = 10/9
let m = 28/27
let s = 81/80
L;s;L;s;L;s;m;s;L;s;L;s;m;s;
stack()
41@
</pre>

{{Cat|JI groups}}

Porcupine

2026-05-22T21:40:23Z

Lériendil: added septendecimal interpretations of septimal porcupine

'''Porcupine''', [15 & 22] (usually defined in 2.3.5.11 or 2.3.5.7.11), is a temperament that splits 4/3 into three submajor seconds (approximately 11/10), representing 10/9~11/10~12/11. In the 5-limit, it equates 81/80 with 25/24. This makes it an excellent compromise between accuracy and simplicity on the side of simplicity while at the same time not fully exotempering (the intervals it detunes significantly, such as 10/9, can be seen as "connective" intervals rather than distinct harmonic identities, with the notable exception of 11/9). Porcupine is also notable for being inherently "un-Meantone" in the sense that rather than tempering out 81/80, it equates 81/80 to a fundamental 5-limit structural interval 25/24, the difference between 5/4 and 6/5; in fact, 7edo (essentially a trivial tuning of both) is the unique edo that is both Porcupine and Meantone.

The simplest Porcupine edo join is [7 & 8], and surprisingly this correctly defines 11-limit porcupine (implying 8edo technically supports [[archy]]) - however this results in an inaccurate extension to higher primes than 11. Also note that 29 agrees with 15 & 22 in 2.3.5.11 but not in 2.3.5.7.11, thus [22 & 29] represents a separate extension from [15 & 22] in the full 11-limit.

The Porcupine generator generates 1L6s, 7L1s, and 7L8s.

== Interval chain ==
In the following table, odd harmonics 1–15 and their inverses are in '''bold'''. Interpretations in parentheses are only found in the Septimal Porcupine (2.3.5.7.11.17 [15 & 22]) extension.
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 162.8
| 10/9, 11/10, 12/11
|-
| 2
| 325.6
| 6/5, 11/9, (17/14)
|-
| 3
| 488.4
| '''4/3'''
|-
| 4
| 651.3
| '''16/11'''
|-
| 5
| 814.1
| '''8/5'''
|-
| 6
| 976.9
| ('''7/4'''), '''16/9'''
|-
| 7
| 1139.7
| 160/81, 64/33, 48/25
|-
| 8
| 102.5
| '''16/15''', ('''17/16''')
|-
| 9
| 265.3
| (7/6)
|-
| 10
| 428.2
| (14/11), 32/25
|-
| 11
| 591.0
| (7/5, 17/12)
|-
| 12
| 753.8
| (14/9, 17/11)
|-
| 13
| 916.6
| (17/10)
|}
<nowiki>*</nowiki> In 2.3.5.7.11 CWE tuning

== Notation and intervals ==
Each sharp or flat from MOS diatonic can be split into three distinct notes, so we use the accidental ^ to raise by a 1/3 chroma and v to lower by a 1/3 chroma (ups and downs notation). Other accidentals that are identified with this in porcupine include any accidental representing the syntonic comma (such as in Ben Johnston, sagittal, SRS, or FJS notation), any accidental representing 25/24 (also Ben Johnston), and any accidental representing 33/32 (such as the FJS or HEJI accidentals for 11).

To avoid ambiguity, systems of notation that utilize the porcupine 25/24 diesis as their chroma may use exclusively ups and downs, though it might be more natural to some to repurpose the diatonic # and b symbols, especially if diatonic notation is not used simultaneously with these other schemes.

Zarlino notation uses the [[ternary]] Zarlino scale (see [[#Zarlino diatonic]]), or Ptolemy's intense diatonic, as its basic scale. Alternatively, the MOS tempering of equable heptatonic may be used, with degrees from 1 through 7 instead of note names.

As porcupine (15 & 22 extension) is a keemic temperament, it has four evenly spaced interval qualities: subminor, nearminor, nearmajor, and supermajor. These are the qualities found in 22edo, but they may also be applied to other porcupine systems such as 15 or 37edo. They may also be used more generally in just intonation, where they are not evenly spaced.

Porcupine also therefore gives a distinction between MOS diatonic (the standard superpyth diatonic, with each chroma split in three parts) and "zarlino" diatonic, wherein zarlino may be seen as a MODMOS of onyx.

== Modal harmony ==
Modal harmony further emphasizes the qualities of the various intervals and chords found in the different scales used in music, as opposed to things like leading tendencies. It is within modal harmony that clear "supermajor", "nearmajor", "nearminor", and "subminor" diatonic scales can be defined, rather than used as context-dependent tonal systems. These mostly follow the interval qualities suggested above, except this time it becomes applicable to an entire scale rather than just to specific chords. (And of course, additional modes of zarlino or mosdiatonic may be used.)
[[File:Modes_in_22edo.png|thumb|252x252px|The modes presented here, arranged in a Tetrahedron.]]
Given this, it's also useful to enumerate various modal scales, as a counterpart to the various non-Ionian/Aeolian modes used throughout standard modal harmony. These will not be exclusively "real" diatonic modes, but rather combinations of qualities loosely analogous to standard modes (and sharing the quality of having notes constrained to range in certain qualities), in six "series" comprising 22 unique modes, visible on the right. This setup overall aims to generalize the idea that diatonic modes exist on a gradation of "brightness" in 12edo, where successive alterations make a mode brighter or darker. Here, bright vs. dark isn't the only axis, however - there's near vs. super/sub and stable vs. unstable as well, so series of alterations along those allow for a much more complex selection of modes to choose from. Additionally, each series of modes has one quality in common, which I've labelled here, so all the "stable" modes have only stable intervals, even if they might contain both nearmajor and subminor ones, and all the "bright" modes contain only major intervals, even if they might be both nearmajor and supermajor. Holding the fourth and fifth constant (as Lydian and Locrian are rarely used in standard diatonic modal harmony) means that there are four "vertex" modes, corresponding to the pure nearmajor, nearminor, supermajor, and subminor qualities, as well as the Ionian and Phrygian modes of Zarlino and mosdiatonic.

==== Choosing a mode ====
Much as the choice of mode in 12edo largely depends on its position on the scale from bright to dark, you might choose a mode here by selecting a series based on the common sound you want your song or section to have, and then choosing a position on that series between its two extremes. For example, for something intense and somewhat uncanny, you might start by choosing the Unstable series, and then proceed to select a mode along that series between bright/super and near/dark that embodies the feel you want, such as Unstable Dorian. Alternatively, for an excited, cheerful sound, you might choose the Bright series and a mode between the near/stable and super/unstable extremes of it, such as Didymic Major.

The "Equable" mode (which is the main porcupine MOS) serves as a somewhat 'neutral' sound - despite the lack of neutral intervals in porcupine, it still occupies that somewhat soft position in between major and minor qualities, while at the same time being more equally distributed than any other version of Dorian available. Quality-wise, it has a mix of nearmajor (bright, stable) and nearminor (dark, unstable) intervals, serving as the opposite polarity to MOS Dorian, and its equidistant nature somewhat overrides other quality-based properties from a melodic perspective. On the opposite side of things, MOS Dorian can be seen as somewhat aggressively defined by its qualities, being a mix of subminor (stable, dark) and supermajor (unstable, bright), with a subminor third on the tonic.

Other pairs of "opposing" modes include unstable Dorian vs. stable Dorian, and didymic major vs. didymic minor, both of which unlike equable vs. MOS Dorian form complementary pairs similar to Ionian and Phrygian in 12edo.

Here they have been organized into two "loops"; bolded entries represent modes that differ along the loops, and italicized entries have had their positions flipped.

All modes are given in 22edo tuning.
{| class="wikitable"
|+Class A
! colspan="4" |Loop A
! colspan="4" |Loop B
|-
!Series
!Mode
!Type
!Name
!Series
!Mode
!Type
!Name
|-
|Super/Sub
|{{Interval ruler|22|0, 200, 250, 500, 700, 760, 1000, 1200}}
|Aeolian
|Aeolian
|Super/Sub
|{{Interval ruler|22|0, 200, 250, 500, 700, 760, 1000, 1200}}
|Aeolian
|Aeolian
|-
|Super/Sub
|{{Interval ruler|22|0, 200, 250, 500, 700, 930, 1000, 1200}}
|Dorian
|Dorian
|Super/Sub
|{{Interval ruler|22|0, 200, 250, 500, 700, 930, 1000, 1200}}
|Dorian
|Dorian
|-
|Super/Sub
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1000, 1200}}
|Mixolydian
|Mixolydian
|Super/Sub
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1000, 1200}}
|Mixolydian
|Mixolydian
|-
|Bright, Super/Sub, Unstable
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1150, 1200}}
|Ionian
|Ionian ("Supermajor")
|Bright, Super/Sub, Unstable
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1150, 1200}}
|Ionian
|Ionian ("Supermajor")
|-
|'''Bright'''
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1100, 1200}}
|Ionian
|'''Harmonic major'''
|'''Unstable'''
|{{Interval ruler|22|0, 220, 430, 500, 700, 930, 1050, 1200}}
|Mixolydian
|'''Unstable Mixolydian'''
|-
|'''Bright'''
|{{Interval ruler|22|0, 200, 380, 500, 700, 930, 1100, 1200}}
|Ionian
|'''Didymic major'''
|'''Unstable'''
|{{Interval ruler|22|0, 220, 330, 500, 700, 930, 1050, 1200}}
|Dorian
|'''Unstable Dorian'''
|-
|'''Bright'''
|{{Interval ruler|22|0, 200, 380, 500, 700, 880, 1100, 1200}}
|Ionian
|'''RH-Ionian'''
|'''Unstable'''
|{{Interval ruler|22|0, 220, 330, 500, 700, 830, 1050, 1200}}
|Aeolian
|'''LH-Aeolian'''
|-
|''Near, Bright, Stable''
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1100, 1200}}
|Ionian
|''LH-Ionian ("Nearmajor")''
|''Near, Dark, Unstable''
|{{Interval ruler|22|0, 100, 330, 500, 700, 830, 1050, 1200}}
|Phrygian
|''RH-Phrygian ("Nearminor")''
|-
|''Near''
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1050, 1200}}
|Mixolydian
|''Major equable''
|''Near''
|{{Interval ruler|22|0, 160, 330, 500, 700, 830, 1050, 1200}}
|Aeolian
|''Minor equable''
|-
|Near
|{{Interval ruler|22|0, 160, 330, 500, 700, 880, 1050, 1200}}
|Dorian
|Equable
|Near
|{{Interval ruler|22|0, 160, 330, 500, 700, 880, 1050, 1200}}
|Dorian
|Equable
|-
|''Near''
|{{Interval ruler|22|0, 160, 330, 500, 700, 830, 1050, 1200}}
|Aeolian
|''Minor equable''
|''Near''
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1050, 1200}}
|Mixolydian
|''Major equable''
|-
|''Near, Dark, Unstable''
|{{Interval ruler|22|0, 100, 330, 500, 700, 830, 1050, 1200}}
|Phrygian
|''RH-Phrygian ("Nearminor")''
|''Near, Bright, Stable''
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1100, 1200}}
|Ionian
|''LH-Ionian ("Nearmajor")''
|-
|'''Dark'''
|{{Interval ruler|22|0, 100, 330, 500, 700, 830, 1000, 1200}}
|Phrygian
|'''LH-Phrygian'''
|'''Stable'''
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 980, 1200}}
|Mixolydian
|'''RH-Mixolydian'''
|-
|'''Dark'''
|{{Interval ruler|22|0, 100, 270, 500, 700, 830, 1000, 1200}}
|Phrygian
|'''Didymic minor'''
|'''Stable'''
|{{Interval ruler|22|0, 160, 270, 500, 700, 880, 980, 1200}}
|Dorian
|'''Stable Dorian'''
|-
|'''Dark'''
|{{Interval ruler|22|0, 100, 270, 500, 700, 770, 1000, 1200}}
|Phrygian
|'''Subharmonic minor'''
|'''Stable'''
|{{Interval ruler|22|0, 160, 270, 500, 700, 760, 980, 1200}}
|Aeolian
|'''Stable Aeolian'''
|-
|Dark, Super/Sub, Stable
|{{Interval ruler|22|0, 50, 270, 500, 700, 770, 1000, 1200}}
|Phrygian
|Phrygian ("Subminor")
|Dark, Super/Sub, Stable
|{{Interval ruler|22|0, 50, 270, 500, 700, 770, 1000, 1200}}
|Phrygian
|Phrygian ("Subminor")
|}
Note that Aeolian is not a vertex. Because of this, it might be prudent to construct a secondary, smaller tetrahedron that holds the major second constant alongside the fourth and fifth. Doing so yields six additional modes:

* A set of two additional modes between RH-Ionian and LH-Aeolian, acting as alternative near forms of Mixolydian/major equable ({{Interval ruler|22|0, 220, 380, 500, 700, 880, 1050, 1200}}) and Dorian/equable ({{Interval ruler|22|0, 220, 330, 500, 700, 880, 1040, 1200}} )

* A set of two additional modes between LH-Aeolian and mosdiatonic Aeolian, acting as alternative dark/minor scales ({{Interval ruler|22|0, 220, 330, 500, 700, 830, 1000, 1200}}, {{Interval ruler|22|0, 220, 270, 500, 700, 830, 1000, 1200}} ).

* Alternative stable forms of Mixolydian ({{Interval ruler|22|0, 200, 380, 500, 700, 880, 980, 1200}}) and Dorian ({{Interval ruler|22|0, 200, 270, 500, 700, 880, 980, 1200}} ). These differ by one note varying by two steps; between them is in fact the simplest possible 5-limit Dorian, at ({{Interval ruler|22|0, 200, 330, 500, 700, 880, 980, 1200}} ), which is not a mode of zarlino due to distributing the large and medium steps differently.

This appears to suggest that the sum total of all theoretically possible modes existing under this system is the complete volume of a tetrahedron with endpoints at near- Locrian and Lydian and at sub-Locrian and super-Lydian. There are 84 total modes in the scheme, which are the rotations of the following 8 base scales, including chirality. These are the set of scales that have the property that all instances of any diatonic interval between any two notes in the scale are either supermajor, nearmajor, nearminor, or subminor, which is the property that constrains the tetrahedron:
{| class="wikitable"
|+
!Name
!Scale
!Note
!Symmetrical?
!Exists in the set of 22 modes?
|-
|mosdiatonic
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1150, 1200}}
|
|No
|Yes
|-
|harmonic major
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1100, 1200}}
|
|Yes
|Yes
|-
|didymic
|{{Interval ruler|22|0, 200, 380, 500, 700, 930, 1100, 1200}}
|
|Yes
|Yes
|-
|zarlino
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1100, 1200}}
|
|Yes
|Yes
|-
|diatonyx-A
|{{Interval ruler|22|0, 220, 380, 500, 700, 880, 1050, 1200}}
|The upper tetrachord is a porcupine tetrachord.
|No
|No
|-
|diatonyx-B
|{{Interval ruler|22|0, 160, 330, 500, 700, 830, 1050, 1200}}
|The lower tetrachord is a porcupine tetrachord.
|Yes
|Yes
|-
|equable / onyx
|{{Interval ruler|22|0, 160, 330, 500, 700, 880, 1050, 1200}}
|
|No
|Yes
|-
|symmetrical dorian
|{{Interval ruler|22|0, 200, 330, 500, 700, 880, 980, 1200}}
|
|No
|No
|}
This reduces to a set of 35 if the fourth and fifth are held fixed, and 55 if only the fifth is.

Every mode of one of these scales has a pattern of broadly major and minor intervals corresponding to one of the standard diatonic modes. For example, the equable scale in its primary mode is a form of Dorian, as its pattern is major-minor-perfect-perfect-major-minor (in this case, nearmajor and nearminor). However, there are not in fact 12 instances of each mode! The equable scale only has Mixolydian, Dorian, and Aeolian modes, and the symmetrical Dorian scale lacks a Locrian or Lydian mode.
{| class="wikitable"
|+
!Mode type
!84-set
!35-set
!22-set
|-
|Locrian
|7
| -
| -
|-
|Phrygian
|12
|5
|5
|-
|Aeolian
|15
|8
|4
|-
|Dorian
|16
|9
|4
|-
|Mixolydian
|15
|8
|4
|-
|Ionian
|12
|5
|5
|-
|Lydian
|7
| -
| -
|}
Regardless, this is simply a mathematically complete enumeration - for actual modal music, it is best to stick to the list of 22 modes provided above, as those are the ones that have clear common qualities alongside the functionally important perfect fifth and fourth.

Additional work needs to be done to determine if this can be generalized.

==== Tetrachords in modal analysis ====
Tetrachords can be used in a different way, more in accordance with their use in modern 12edo theory. In modes where the fourth and fifth are perfect, the mode can always be thought of as being comprised of two tetrachords separated by a whole tone, although the constraints on these tetrachords are entirely different from the Greek versions. In short, a modal tetrachord must comprise the unison, the fourth, a second of one of the four qualities, and a third of one of the four qualities, such that the interval between two adjacent tones is never more than four steps. This is a generalization of the constraints on modal tetrachord patterns in 12edo, which must always contain either whole tones or semitones. By this constraint there are ten distinct tetrachords in 22edo. Considering all the possible scales constructed from these, there are 10x10 = 100 distinct possibilities, compared to the 3x3 = 9 options found in 12edo. This provides an extended set, including not only the 35 modes corresponding to diatonic but 65 additional scales corresponding in some regard to melodic minor or neapolitan major. Not all intervals are necessarily within their expected quality ranges.

Loosening the constraint further to only necessitate that the two movable tones remain within the 4 qualities of their respective degrees allows for the generalization to a set of scales analogous to harmonic minor or double harmonic major, with 156 additional possibilities.

=== Chromatic subsets ===
In porcupine, multiple qualities may be combined together into a compound system. With mosdiatonic alone, there is little reason to do this, because there are only two qualities available, so the scale combining them (the chromatic scale, or some other large scale like {{Interval ruler|12|0, 200, 300, 400, 500, 700, 800, 900, 1100, 1200}} (12edo tuning)) is not particularly engaging from either a tonal or modal perspective. However, in porcupine there are four different qualities, from which two may be selected to share characteristics.

The standard chromatic scales combine nearmajor+nearminor and supermajor+subminor, which lead to somewhat of the same problem as 12edo chromatic; they are opposing pairs of qualities. However, if we make an ''asymmetric'' chromatic scale, with (for instance) supermajor and nearminor, we get a scale with the trait they have in common: being "unstable". Alternatively, you could get a generally "dark" system by combining subminor and nearminor qualities.

The following are a few examples of these kinds of scales, including diatonic and chromatic variations. (Note that in tonal music, these become less distinct from standard counterparts, as degrees are already expected to be altered between different qualities depending on context.)

Aberrismic scales may also be leveraged for this purpose.

== Functional harmony ==
See [[Archy#Full 7-limit harmony]].

=== Harmony involving the 1/3-chroma equivalence ===

== Scales ==

=== Zarlino ===
The 5-limit zarlino scale, in Porcupine temperament, has its two chromas 25/24 and 81/80 equated, making it a [[MODMOS]] of onyx. This characteristic defines 5-limit Porcupine. It allows zarlino to be notated with a single pair of accidentals, which may be written as ups and downs if MOSdiatonic sharps are used, consistently with the general usage of ups and downs in porcupine.

Including degrees of both zarlino and Pythagorean diatonic results in [[blackdye]]. In fact, blackdye or several characteristics of it are likely to naturally emerge in tonal harmony in the first place, given an overall desire to avoid wolf fifths in nearmajor and nearminor tonalities, resulting in certain intervals being doubled up. Most notably, it is reasonable to consider the two forms of major second equally part of a nearmajor tonal system, so that 5-2 and 2-6 can both be perfect fifths with different versions of the 2 degree (although unless 6 is additionally sharpened, some additional harmonic movements are needed to resolve the edostep offset that results from a pumped syntonic comma if you move from fifth-bounded triads on 5 to 2 in a single motion).

Zarlino may be used as the base scale for porcupine. This encounters problems with existing familiarity with notation (for example, one of D-G and G-C must now be a flat 5th, called a "wolf fifth" and representing 16/11 as opposed to 3/2), but may be more structurally useful than considering equiheptatonic the base scale.

==== Equiheptatonic ====
The equable diatonic, 1/(18:20:22:24:27:30:33:36) (or in this case, equivalently its [[otonal]] counterpart) is represented as the MOS scale sssLsss in porcupine. It is reasonable to, for that structural reason, consider sssLsss the default mode, with a nearminor chord on the tonic - it is the unique mode which possesses both a perfect fifth and a perfect fourth. This is more generally the MOS porcupine[7]; altering several notes of this MOS yields the Zarlino diatonic, explaining 22edo Zarlino's heavy reliance on porcupine's equivalences. Porcupine also has an 8-note scale LLLLsLLL that adds an additional "blue note" to the heptatonic, and a chromatic scale sLsLsLsLsLsLsLs, a form of the Roklotian scale that may also be derived by dividing the intervals of superpyth pentatonic: sLsLs -> [sLs] [LsL] [sLs] [LsL] [sLs].
{| class="wikitable"
|+
!Mode
!Brightness
!Steps
!Interval qualities
|-
|Mixolydian
|3
|Lssssss
|AMMMMP
|-
|Mixolydian
|2
|sLsssss
|PMMMMP
|-
|Dorian
|1
|ssLssss
|PmMMMP
|-
|Dorian
|0
|sssLsss
|PmmMMP
|-
|Dorian
| -1
|ssssLss
|PmmmMP
|-
|Aeolian
| -2
|sssssLs
|PmmmmP
|-
|Aeolian
| -3
|ssssssL
|Pmmmmd
|}

== Solfege ==
A solfege provided by Vector for porcupine uses an -n coda to inflect inwards by a 1/3-chroma; -s may be used to inflect outwards by a 1/3-chroma. The standard 12-form solfege syllables (do, ra, re, me, mi...) represent Superpyth[12].

An alternate solfege proposed by Kite consistently uses the sequence -i, -u, -o, and -a for the interval qualities.

A solfege used by Andrew Heathwaite uses the following syllables:
{| class="wikitable"
!Note
!Solfege (on Do)
!Note
!Solfege (on Do)
|-
|P1 / P8
|'''Do'''
|vA4
|Fi
|-
|m2
|Di
|vP5
|Su
|-
|^m2
|Ra
|P5
|'''Sol'''
|-
|vM2
|Ru
|m6
|Lo
|-
|M2
|'''Re'''
|^m6
|Le
|-
|m3
|Ma
|vM6
|'''La'''
|-
|^m3
|Me
|M6
|Li
|-
|vM3
|'''Mi'''
|m7
|Ta
|-
|M3
|Mo
|^m7
|Tu
|-
|P4
|'''Fa'''
|vM7
|'''Ti'''
|-
|^P4
|Fu
|M7
|Da
|}

== List of patent vals ==
{| class="wikitable"
|+
!EDO
!Extension to 7
!Generator tuning
!25/24 tuning
!Fifth tuning
|-
|7
|[15 & 22]
|171.4c
|0c
|685.7c
|-
|29
|[22 & 29]
|165.5c
|41.4c
|703.5c
|-
|51
|[22 & 29]
|164.7c
|47.1c
|705.9c
|-
|22
|[15 & 22], [22 & 29]
|163.6c
|54.5c
|709.1c
|-
|59
|[15 & 22]
|162.7c
|61c
|711.9c
|-
|37
|[15 & 22]
|162.2c
|64.9c
|713.5c
|-
|15
|[15 & 22]
|160c
|80c
|720c
|-
|8
|[15 & 22]
|150c
|150c
|750c
|}

{{Navbox regtemp}}
{{Cat|temperaments}}

Talk:Porcupine

2026-05-22T19:21:05Z

Lériendil: Created page with "== Content moved from 22edo? == I'd surmise if a lot of 22edo diatonic harmony content got moved here, not all of it belongs appropriately on the Porcupine page compared to someplace else. --~~~~"

== Content moved from 22edo? ==
I'd surmise if a lot of 22edo diatonic harmony content got moved here, not all of it belongs appropriately on the Porcupine page compared to someplace else. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 19:21, 22 May 2026 (UTC)

User talk:Kaiveran

2026-05-22T14:44:22Z

Lériendil:

== Do not add scala files ==

Do not add scala files to XenReference. (Also, do not add entire pages just for single scales with no theory info.) (Also also, title pages in a readable manner.)

-- [[User:Vector|Vector]] ([[User talk:Vector|talk]]) 18:11, 18 May 2026 (UTC)

: Apologies. I was not sober and somehow the use of the humble text editor to urgently save a ''text'' file didn't occur to me. Is "no Scala files" a blanket policy? If so, I wouldn't even include it as a 31EDO subpage, as the most recent edit has done, and just nominate it for deletion. I plan to work on a page discussing this and other partial skip-fretting schemes in detail (where these scala files are used to physically model the fretting scheme, hence the previously included link.) [[User:Kaiveran|Kaiveran]] ([[User talk:Kaiveran|talk]]) 06:48, 22 May 2026 (UTC)

:: On my part, I'd say that if you're using a Scala file as part of a larger article you're assembling, it's best practice to put it in a user page sandbox. I'm reluctant to make blanket policies for basically anything though; though "entire pages just for single scales with no theory info" is going to be discouraged as we don't want to proliferate anything that's pure documentation without additional substance to it. Ideally, Scala files as part of pages that list different scales and have some explanation attached to them would be best practice; though in the modern day SW2 or SW3 links might be preferable anyhow. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 14:44, 22 May 2026 (UTC)

User talk:Kaiveran

2026-05-22T14:40:39Z

Lériendil: Undo revision 7284 by Lériendil (talk)

User talk:Kaiveran

2026-05-22T14:40:15Z

Lériendil: Reverted edit by Kaiveran (talk) to last revision by Vector

4/3

2026-05-21T08:05:16Z

Lériendil: Redirected page to Perfect fourth

#redirect [[Perfect fourth]]

22edo

2026-05-21T06:04:41Z

Lériendil:

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (as in the [[ADIN]] system for melodic qualities, which will be used in much of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (which can be called "nearminor" and "nearmajor" to disambiguate). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma superpyth.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and 11/9, or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and 7/5).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''Sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "Father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''Orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remembering that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equitetrachordal diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualized as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|Magic may be conceptualized decatonically as well; however, 4/3 and 3/2 are placed on the same degree, unlike in a standard 10-form.
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T06:01:12Z

Lériendil: /* Tables of scales */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma superpyth.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and 11/9, or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and 7/5).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''Sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "Father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''Orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remembering that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equitetrachordal diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualized as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|Magic may be conceptualized decatonically as well; however, 4/3 and 3/2 are placed on the same degree, unlike in a standard 10-form.
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T05:59:52Z

Lériendil: /* Building scales from tetrachords */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma superpyth.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and 11/9, or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and 7/5).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''Sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "Father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''Orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remembering that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equitetrachordal diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T05:59:25Z

Lériendil: /* Building scales from tetrachords */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma superpyth.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and 11/9, or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and 7/5).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''Sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "Father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''Orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remembering that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T05:59:03Z

Lériendil: /* 11edo temperaments */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma superpyth.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and 11/9, or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and 7/5).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''Sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "Father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''Orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T05:58:31Z

Lériendil: /* Edostep interpretations */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma superpyth.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and 11/9, or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and 7/5).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T05:58:09Z

Lériendil: /* Edostep interpretations */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma superpyth.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and 7/5).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T05:57:28Z

Lériendil: /* JI approximation */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma superpyth.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T05:57:03Z

Lériendil: /* JI approximation */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma Archy tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T05:56:39Z

Lériendil: removed repeated links

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by Porcupine (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: pajara and diatonic (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically Pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T04:41:39Z

Lériendil: /* Compositional theory */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting [[Porcupine]], the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T04:41:28Z

Lériendil: /* Compositional theory */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting [[porcupine]], the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T04:37:59Z

Lériendil: preliminarily adding the /Scales content back to the main article

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

==== Tables of scales ====
The following is a table of scales in 22edo.

===== Porcupine scales =====
MOS scales generated by a nearmajor second.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}
|The same as the "equable Dorian" discussed above.
|-
|Pine
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}
|
|-
|Roklotic
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.
|}

===== Orwell scales =====
MOS scales generated by a subminor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Manual
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.
|-
|Gramitonic
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.
|-
|Antiparagonic
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.
|}

===== Magic scales =====
MOS scales generated by a nearmajor third.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Mosh
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.
|-
|Sephiroid
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}
|
|-
|Antiluachoid
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}
|
|}

===== Superpyth scales =====
MOS scales generated by a perfect fifth.
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Pentic
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".
|-
|Mosdiatonic
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.
|-
|P-chromatic
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}
|
|}

===== Half-octave scales =====
MOS scales generated against the half-octave.
{| class="wikitable"
!Temperament
!Name
!Chart
!Notes
|-
| rowspan="2" |Pajara
|jaric
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|
|-
|telluric
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.
|-
| rowspan="3" |Hedgehog
|malic
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}
|One of three tunings of malic available in 22edo.
|-
|ekic
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}
|
|-
| -
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}
|
|-
| rowspan="2" |Astrology
|citric
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}
|One of two tunings of citric available in 22edo.
|-
|lemon
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}
|
|-
| rowspan="2" |Doublewide
|citric
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.
|-
|lime
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}
|
|}

===== Additional scales =====
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Zarlino
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}
|The 5-limit diatonic in 22edo.
|-
|Pentachordal pajara
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|
|-
|Tellurian
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}
|
|}

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo/Scales

2026-05-21T04:37:58Z

Lériendil: preliminarily adding the /Scales content back to the main article

#redirect [[22edo #Tables of scales]]

22edo

2026-05-21T04:36:16Z

Lériendil: preliminarily adding the /Chords content back to the main article

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

=== Tables of chords ===
The following is a table of chords in 22edo.

''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''

==== Fifth-bounded tertian triads ====
Three-note chords built out of thirds, bounded by a perfect fifth.
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
|-
|supermajor (M)
|supermajor
|perfect
|[0 8 13]
|-
|nearmajor (P, unmarked)
|nearmajor
|perfect
|[0 7 13]
|-
|nearminor (p)
|nearminor
|perfect
|[0 6 13]
|-
|subminor (m)
|subminor
|perfect
|[0 5 13]
|}

==== Other tertian triads ====
Additional three-note chords built out of thirds.

===== Augmented triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near augmented (z+)
|nearmajor
|up
|[0 7 14]
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.
|-
|exo augmented (S+)
|supermajor
|augmented
|[0 8 16]
|"Neutral" counterpart of 5/3-bounded chords.
|}

===== Diminished triads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Edostep
!Notes
|-
|near diminished (z°)
|nearminor
|down
|[0 6 12]
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.
|-
|major diminished (°)
|nearminor
|updiminished (tritone)
|[0 6 11]
|5:6:7. Found in harmonic 4:5:6:7.
|-
|minor diminished (m°)
|subminor
|updiminished (tritone)
|[0 5 11]
|
|-
|exo diminished (S°)
|subminor
|diminished
|[0 5 10]
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.
|}

==== Tetrads ====

===== Supermajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|exodominant seventh (S7)
|supermajor
|perfect
|subminor
|10
|[0 8 13 18]
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.
|-
|supermajor seventh (M7, Δ7)
|supermajor
|perfect
|supermajor
|13 (P5)
|[0 8 13 21]
|Seventh chord of supermajor.
|-
|supermajor nearmajor seventh (MP7)
|supermajor
|perfect
|nearmajor
|12
|[0 8 13 20]
|Acts as a more directed version of a M7 chord.
|}

===== Nearmajor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|harmonic seventh (7), major harmonic (H)
|nearmajor
|perfect
|subminor
|11 (tritone)
|[0 7 13 18]
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).
|-
|neardominant seventh (z7)
|nearmajor
|perfect
|nearminor
|12
|[0 7 13 19]
|
|-
|nearmajor seventh (P7)
|nearmajor
|perfect
|nearmajor
|13 (P5)
|[0 7 13 20]
|Seventh chord of nearmajor.
|-
|nearmajor supermajor seventh (PM7)
|nearmajor
|perfect
|supermajor
|14
|[0 7 13 21]1]
|Acts as a less directed version of a P7 chord.
|}

===== Nearminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|minor harmonic (Hm)
|nearminor
|perfect
|supermajor 6th
|11 (tritone)
|[0 6 13 17]
|
|-
|nearminor seventh (p7)
|nearminor
|perfect
|nearminor
|13 (P5)
|[0 6 13 19]
|Seventh chord of nearminor.
|-
|nearminor nearmajor seventh (pP7)
|nearminor
|perfect
|nearmajor
|14
|[0 6 13 20]
|Seventh chord of harmonic nearminor.
|-
|nearminor subminor seventh (pm7)
|nearminor
|perfect
|subminor
|12
|[0 6 13 18]
|
|}

===== Subminor tetrads =====
{| class="wikitable"
!Name
!Third
!Fifth
!Seventh
!Steps between 3 and 7
!Edostep
!Notes
|-
|subminor seventh (m7)
|subminor
|perfect
|subminor
|13 (P5)
|[0 5 13 18]
|Seventh chord of subminor.
|-
|subminor nearminor seventh (mp7)
|subminor
|perfect
|nearminor
|14
|[0 5 13 19]
|
|-
|subminor nearmajor seventh (mP7)
|subminor
|perfect
|nearmajor
|15
|[0 5 13 20]
|Seventh chord of harmonic subminor.
|}

==== Non-tertian functional chords ====
{| class="wikitable"
!Name
!Mediant
!Bounding interval
!Edostep
!Notes
|-
|chthonic minor (Lm)
|minor unilatus (whole tone)
|perfect fourth
|[0 4 9]
|
|-
|chthonic major (LM)
|major unilatus (subminor third)
|perfect fourth
|[0 5 9]
|6:7:8 chord.
|-
|suspended 4th (sus4)
|perfect 4th
|perfect fifth
|[0 9 13]
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.
|-
|suspended up4th (sus^4)
|up 4th
|perfect fifth
|[0 10 13]
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.
|-
|suspended 2nd (sus2)
|supermajor 2nd
|perfect fifth
|[0 4 13]
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.
|-
|suspended down2nd (susv2)
|nearmajor 2nd
|perfect fifth
|[0 3 13]
|Suspension resolves to subminor
|-
|naiadic minor (S+m)
|nearmajor third
|nearmajor sixth
|[0 7 16]
|
|-
|naiadic major (S+M)
|perfect fourth
|nearmajor sixth
|[0 9 16]
|3:4:5 chord.
|}

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo/Chords

2026-05-21T04:36:15Z

Lériendil: preliminarily adding the /Chords content back to the main article

#redirect [[22edo #Tables of chords]]

40edo

2026-05-21T03:08:25Z

Lériendil: /* Tempered commas */

[[File:40edo_esstemp.png|thumb|The 2.5.7/3.11/3 Guanyintet chain is central to the structure of 40edo; additionally, harmonies of 13 and 19 such as 10:13:16:19 can be reached in several ways.]]
'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group, though for structural reasons it may be better to include the 3 regardless (and thus to use the patent val).

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s (sLsLsLsLs) on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]], S11/S14 (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three [[20/19]]'s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several 10:13:16:19 tetrads available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament. While the whole tone in Deeptone approximates 10/9 very well, it cannot be interpreted that way outside of the dual-prime interpretation.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of Deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of nearmajor, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell/Guanyintet ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes. This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

An alternative to Orwell[13] worth mentioning is "[https://scaleworkshop.plainsound.org/scale/pwxnuq0Gb Orwell[14]]", constructed by splitting, rather than the large step of Orwell[9], the small step into a 1\40 chroma and a remainder. This can be considered an [[aberrismic]] superset of Orwell[9], and due to the 5:4 hardness of Orwell[9], consists of step sizes 5\, 3\, and 1\40 that all differ by the same amount, 2\40. This is a subset of Orwell[22] that manages to reach the higher harmonies found in the Guanyintet chain.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

==== 13edo-derived muddles ====
A notable scale of 40edo is generated by 3\40, stacking thrice to the Orwell generator of 7/6 and nine times to 8/5. The principal generated [[MOS]] is of 13 steps - twelve being length 3\40 and one being length 4\40, and serving as a [[well-temperament]] of [[13edo]] - and contains multiple instances of the 10:13:16:19 tetrad. As this MOS scale is quite nearly an EDO, one is incentivized to take further subsets of it that have useful melodic and harmonic properties - that is, [[MOS muddle]]s.

As taking every 3 generators essentially gives Orwell, the most notable generator chains of 13edo reflected in 40edo in this manner include the stacks of 4\13 and 5\13, representing [[3L 4s]]/[[3L 7s]], and [[oneirotonic]] respectively. [https://scaleworkshop.plainsound.org/scale/_dyrsuMpS The former sequence] is an easy approach to incorporating 10:13:16:19 into a musically coherent scale, as it occurs even in the 7-note muddle if the 4\40 step is placed correctly. The latter provides a simulation of the oneirotonic scale in the largest EDO to formally lack one.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T02:44:59Z

Lériendil: /* Scale theory */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo/Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

A table of chords may be found at [[22edo/Chords]].

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T02:44:38Z

Lériendil: Undo revision 7258 by Lériendil (talk)

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo#Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

A table of chords may be found at [[22edo/Chords]].

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T02:44:21Z

Lériendil: /* Harmony */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo#Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

=== Chords and harmony ===
==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

A table of chords may be found at [[22edo/Chords]].

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T02:43:12Z

Lériendil: /* Generator sequences */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo#Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

A table of chords may be found at [[22edo/Chords]].

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T02:41:44Z

Lériendil: added link to scaleworkshop

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo#Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo.

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

A table of chords may be found at [[22edo/Chords]].

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T02:35:39Z

Lériendil: /* Scales */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
==== Scale theory ====
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo#Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce a similar 8-note scale to the original, but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo.

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

A table of chords may be found at [[22edo/Chords]].

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T02:35:05Z

Lériendil: /* Notation systems and a table of intervals */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|Pajara]] and [[Porcupine#Notation and intervals|Porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo#Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce a similar 8-note scale to the original, but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo.

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

A table of chords may be found at [[22edo/Chords]].

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

22edo

2026-05-21T02:21:17Z

Lériendil: /* Scale theory */

[[File:Etj.png|thumb|307x307px|22edo visualization]]
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.

22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (including in the [[ADIN]] system for melodic qualities, which will be used in the remainder of this article).

As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as [[porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.

== General theory ==
=== Derivation of 22edo ===
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.

Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.

From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma [[archy]] tuning.

22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).

Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
{{Harmonics in ED|22|31|0}}

=== Intervals of 22edo ===
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.

==== Edostep interpretations ====
22edo's edostep has the following interpretations in the 7-limit:
* 25/24 (the difference between 5/4 and 6/5)
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 81/80 (the difference between [[10/9]] and 9/8)

Including prime 11, it additionally serves as:
* 22/21 (the difference between 7/6 and [[11/9]], or [[14/11]] and 4/3)
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and [[7/5]]).
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]

==== Notation systems and a table of intervals ====
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).

The native-fifths or [[diatonic notation]] system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.

22edo also supports any notation system for [[Pajara#Notation|pajara]] and [[Porcupine#Notation and intervals|porcupine]].

ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the [[Porcupine]] page.

JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
{| class="wikitable"
|+
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category (ADIN)
|-
! rowspan="1" | Native-fifths (ups & downs) !! rowspan="1" | Blackdye/Zarlino (Vector) !! rowspan="1" | Pajara decatonic
|-
|0
|0
|1/1
|C
|C
|0
|Perfect unison
|-
|1
|54.5
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109.1
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|163.6
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218.2
|8/7, '''9/8''', [17/15]
|D
|D#
|2
|(Super)major second
|-
|5
|272.7
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327.3
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|381.8
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436.4
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|490.9
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545.5
|'''11/8''', 15/11
|^F, Gb
|F#
|4#
|Near fourth
|-
|11
|600
|7/5, 10/7, [17/12]
|vF#, ^Gb
|Gbb / Fx
|5
|Tritone
|-
|12
|654.5
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709.1
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|763.6
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818.2
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|872.7
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927.3
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|981.8
|'''7/4''', 16/9, [30/17]
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036.4
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1090.9
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145.5
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|0
|Octave
|}

=== Solfege ===
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by the 11-limit of 22et include:
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11

[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}

=== Arithmetic progressions ===
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.

{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]
=== Notable structural chains ===
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).

3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''[[Porcupine]]''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.

5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.

7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].

Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.

22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].

In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]

==== 11edo temperaments ====
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.

Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)

Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.

== Compositional theory ==
=== Tertian structure ===
22edo is a [[keemic]] temperament, with four distinct types of thirds and in general four distinct interval qualities, as a result of supporting [[porcupine]].

{| class="wikitable"
|+Thirds in 22edo
!Quality
|'''Subminor'''
|Nearminor
|Nearmajor
|'''Supermajor'''
|-
!Cents
|'''273'''
|327
|382
|'''436'''
|-
!Just interpretation
|'''7/6''' (+5.9{{c}})
|6/5 (+11.6{{c}})
|5/4 (-4.5{{c}})
|'''9/7''' (+1.3{{c}})
|}
Diatonic thirds are bolded.

=== Scales ===
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).

One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.

22edo supports the various heptatonic scales supported by [[porcupine]] (see [[Porcupine#Scales]]) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.

It also supports the [[Pajara|Pajara[10]]] scale, which evenly divides each step of the MOS pentatonic scale.

More scales may be found at [[22edo#Scales]].

==== Generator sequences ====
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce a similar 8-note scale to the original, but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo.

Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.

==== Building scales from tetrachords ====
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at [[Tetrachord]].

There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).

===== Other polychordal structures =====
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]

=== Harmony ===

==== Consonance and dissonance properties ====
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.

An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.

An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).

==== Modal and functional harmony. ====
There are two distinct approaches to harmony in 22edo: [[pajara]] and diatonic ([[porcupine]]) (more info found on their respective pages), each with tonal and modal approaches.

In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.

However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.

A table of chords may be found at [[22edo/Chords]].

== Isomorphic layouts and other instrument designs ==
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.

The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}

== Supersets and subsets ==
=== 44edo ===
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
{{Harmonics in ED|44|31|0}}

== Comparisons to other tuning systems ==
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
* [[27edo]] shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.

== Music in 22edo ==
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]

== See also ==
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]
* [[22edo/Chords]]
* [[22edo/Scales]]
* [[22edo/Intervals]]
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector

{{Navbox EDO}}
{{Cat|Edos}}

Odd-limit

2026-05-20T19:52:19Z

Lériendil:

An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-odd-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

It may also be useful to restrict an odd-limit further to within a specific [[prime limit]] or [[subgroup]], and these restrictions are constructed from fractions amongst those odd numbers that are both below the odd-limit and composed of primes in the subgroup: e.g. the [[7-limit|7-prime-limited]] 21-odd-limit comprises intervals of odds 3, 5, 7, 9, 15, and 21; and the [[2.3.5.11 subgroup]] 15-odd-limit comprises intervals of odds 3, 5, 9, 11, and 15. The set of intervals constructed as fractions among an arbitrary set of odds, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond''', and tonality diamonds generalize the idea of odd-limits.

'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

== Distinction and monotonicity ==
Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''2/(''n''2 - 1) (though certain odd-limits can have a gap of size (''n''2 + 1)/''n''2 instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotone in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotone''' in the ''n''-odd-limit.

== Individual odd-limits ==

* [[Perfect consonance|3-odd-limit]]

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

* [[5-odd-limit]]

The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

* [[7-odd-limit]]
* [[9-odd-limit]]

The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9, 9/8, and 9/7 consonant. These may be considered "secondary" consonances.

* [[11-odd-limit]]

More info is on each individual odd-limit page.

Odd-limit

2026-05-20T19:45:37Z

Lériendil:

An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

It may also be useful to restrict an odd-limit further to within a specific [[prime limit]] or [[subgroup]], and these restrictions are constructed from fractions amongst those odd numbers that are both below the odd-limit and composed of primes in the subgroup: e.g. the [[7-limit|7-prime-limited]] 21-odd-limit comprises intervals of odds 3, 5, 7, 9, 15, and 21; and the [[2.3.5.11 subgroup]] 15-odd-limit comprises intervals of odds 3, 5, 9, 11, and 15. The set of intervals constructed as fractions among an arbitrary set of odds, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond''', and tonality diamonds generalize the idea of odd-limits.

'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

== Distinction and monotonicity ==
Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''2/(''n''2 - 1) (though certain odd-limits can have a gap of size (''n''2 + 1)/''n''2 instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotone in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotone''' in the ''n''-odd-limit.

== Individual odd-limits ==

* [[Perfect consonance|3-odd-limit]]

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

* [[5-odd-limit]]

The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

* [[7-odd-limit]]
* [[9-odd-limit]]

The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9, 9/8, and 9/7 consonant. These may be considered "secondary" consonances.

* [[11-odd-limit]]

More info is on each individual odd-limit page.

Odd-limit

2026-05-20T19:45:26Z

Lériendil: restricted odd-limits

An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

It may also be useful to restrict an odd-limit further to within a specific [[prime limit]] or [[subgroup]], and these restrictions are constructed from fractions amongst those odd numbers that are both below the odd-limit and composed of primes in the subgroup: e.g. the [[7-limit|7-limited]] 21-odd-limit comprises intervals of odds 3, 5, 7, 9, 15, and 21; and the [[2.3.5.11 subgroup]] 15-odd-limit comprises intervals of odds 3, 5, 9, 11, and 15. The set of intervals constructed as fractions among an arbitrary set of odds, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond''', and tonality diamonds generalize the idea of odd-limits.

'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

== Distinction and monotonicity ==
Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''2/(''n''2 - 1) (though certain odd-limits can have a gap of size (''n''2 + 1)/''n''2 instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotone in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotone''' in the ''n''-odd-limit.

== Individual odd-limits ==

* [[Perfect consonance|3-odd-limit]]

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

* [[5-odd-limit]]

The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

* [[7-odd-limit]]
* [[9-odd-limit]]

The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9, 9/8, and 9/7 consonant. These may be considered "secondary" consonances.

* [[11-odd-limit]]

More info is on each individual odd-limit page.

Odd-limit

2026-05-20T19:32:21Z

Lériendil:

An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7. The set of intervals comprising an odd-limit, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond'''.

'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

== Distinction and monotonicity ==
Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''2/(''n''2 - 1) (though certain odd-limits can have a gap of size (''n''2 + 1)/''n''2 instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotone in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotone''' in the ''n''-odd-limit.

== Individual odd-limits ==

* [[Perfect consonance|3-odd-limit]]

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

* [[5-odd-limit]]

The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

* [[7-odd-limit]]
* [[9-odd-limit]]

The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9, 9/8, and 9/7 consonant. These may be considered "secondary" consonances.

* [[11-odd-limit]]

More info is on each individual odd-limit page.

Odd-limit

2026-05-20T19:32:10Z

Lériendil: Reverted edit by Lériendil (talk) to last revision by Inthar

An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

== Distinction and monotonicity ==
Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''2/(''n''2 - 1) (though certain odd-limits can have a gap of size (''n''2 + 1)/''n''2 instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotone in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotone''' in the ''n''-odd-limit.

== Individual odd-limits ==

* [[Perfect consonance|3-odd-limit]]

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

* [[5-odd-limit]]

The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

* [[7-odd-limit]]
* [[9-odd-limit]]

The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9, 9/8, and 9/7 consonant. These may be considered "secondary" consonances.

* [[11-odd-limit]]

More info is on each individual odd-limit page.

Odd-limit

2026-05-20T19:31:40Z

Lériendil: diamond...

An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7. The set of intervals comprising an odd-limit, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond'''.

'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

== Distinction and monotonicity ==
Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''2/(''n''2 - 1) (though certain odd-limits can have a gap of size (''n''2 + 1)/''n''2 instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotone in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotone''' in the ''n''-odd-limit.

== Individual odd-limits ==

* [[Perfect consonance|3-odd-limit]]

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

* [[5-odd-limit]]

The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

* [[7-odd-limit]]
* [[9-odd-limit]]

The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9 and 9/8 consonant. These may be considered "secondary" consonances.

* [[11-odd-limit]]

More info is on each individual odd-limit page.

Odd-limit

2026-05-20T19:29:51Z

Lériendil: /* Distinction and monotonicity */

An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

== Distinction and monotonicity ==
Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''2/(''n''2 - 1) (though certain odd-limits can have a gap of size (''n''2 + 1)/''n''2 instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotone in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotone''' in the ''n''-odd-limit.

== Individual odd-limits ==

* [[Perfect consonance|3-odd-limit]]

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

* [[5-odd-limit]]

The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

* [[7-odd-limit]]
* [[9-odd-limit]]

The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9 and 9/8 consonant. These may be considered "secondary" consonances.

* [[11-odd-limit]]

More info is on each individual odd-limit page.

Odd-limit

2026-05-20T19:26:04Z

Lériendil: /* Distinction and monotonicity */

An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

== Distinction and monotonicity ==
Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''2/(''n''2 - 1) (though certain odd-limits can have a gap of size (''n''2 + 1)/''n''2 instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotonic in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotonic''' in the ''n''-odd-limit.

== Individual odd-limits ==

* [[Perfect consonance|3-odd-limit]]

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

* [[5-odd-limit]]

The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

* [[7-odd-limit]]
* [[9-odd-limit]]

The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9 and 9/8 consonant. These may be considered "secondary" consonances.

* [[11-odd-limit]]

More info is on each individual odd-limit page.

Odd-limit

2026-05-20T19:25:18Z

Lériendil: added words about distinction, monotonicity, etc.

An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

== Distinction and monotonicity ==
Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''2/(''n''2 - 1) (though certain odd-limits can have a gap of size (''n''2 + 1)/''n''2 instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotonic in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, for intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it would still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotonic''' in the ''n''-odd-limit.

== Individual odd-limits ==

* [[Perfect consonance|3-odd-limit]]

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

* [[5-odd-limit]]

The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

* [[7-odd-limit]]
* [[9-odd-limit]]

The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9 and 9/8 consonant. These may be considered "secondary" consonances.

* [[11-odd-limit]]

More info is on each individual odd-limit page.