Xenharmonic Reference - User contributions [en]

31edo

2026-05-25T03:07:17Z

Hkm: /* JI approximation */

[[File:31edo whr.png|thumb|418x418px|31edo supports Valentine and Miracle, alongside supporting Meantone.]]
'''31edo''', or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 39 [[cent]]s. It is most commonly known as a tuning of [[Meantone]], and for its accurate approximation of the 2.5.7 [[subgroup]].

31edo as a whole contains a diverse palette of interval qualities and structures ranging from the very familiar to the quite exotic, and remarkably, almost all of these still have a reasonably simple harmonic interpretation. As a meantone system, 31edo's diatonic scale includes the basic qualities of the [[5-limit]], such as the [[perfect fourth]] and [[perfect fifth|fifth]], and the classical minor and major thirds ([[6/5]] and [[5/4]]). But 31edo also includes subminor and supermajor intervals, identifiable with [[septal]] ratios such as [[7/6]] and [[9/7]], and [[neutral]] intervals, identifiable with [[11-limit]] ratios such as [[11/9]].

In terms of structures, or ways of organizing harmony, it should first be noted that 31edo's perfect fifth, of 18 steps, is quite divisible. The fifth can be split in two, giving us a neutral-third temperament, known in this case as [[Mohajira]], which emphasizes heptatonic structure, the 11th harmonic, and 2.3.5.11. Splitting the fifth in three gives us [[Slendric]] (or in this case [[Mothra]]), formed by stacking [[8/7]], and which emphasizes pentatonic structure and 2.3.7. Combining these gives us [[Miracle]], while splitting Slendric into three again gives us [[Valentine]]. In 31edo, each of these provides xenharmonic ways of accessing the 11-limit with more simplicity than Meantone. Yet another way to encompass the 11-limit is given by [[Orwell]], generated by 31edo's subminor third; and of course, as a prime EDO, 31edo contains several more structures unique to itself.

== Theory ==
==== Edostep interpretations ====
31edo's edostep has the following interpretations in the 11-limit:

* 128/125 (the residue between three stacked 5/4s and the octave)
* 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)
* 50/49 (the difference between [[7/5]] and [[10/7]])
* 64/63 (the difference between 8/7 and [[9/8]])
* 33/32 (the difference between [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or [[11/10]] and 9/8)
* 55/54 (the difference between 6/5 and 11/9, or 12/11 and [[10/9]])
* 56/55 (the difference between 5/4 and [[14/11]])

==== JI approximation ====
31edo can be understood as a 7-limit system with a somewhat flat 3/2 but nearly-perfect 5th and 7th harmonics. In particular, the product of 5 and 7, [[35/32]], is approximated to within about 0.3{{c}}. 31edo also has an approximation to the 11th harmonic that, while tuned flat, has the property that the flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral third. The harmonic 23 turns out to be flat in a very similar way to 11.

The intervening harmonics - 13, 17, and 19 - are tuned rather sharp, but by almost exactly the same amount; therefore the chord 13:17:19 is extremely well-approximated by 31edo, with the interval [[17/13]] tuned less than 0.1{{c}} off.

31edo's fifth generates a functional diatonic scale. Its whole tone, of 5 steps, is split into semitones of 2 and 3; as 31edo's fifth is flatter than that of [[12edo]], the chromatic semitone, comprised by 2 steps, is smaller than the diatonic semitone, which is 3 steps.
{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality ([[ADIN]])
|Subminor
|'''Nearminor'''
|Neutral
|'''Nearmajor'''
|Supermajor
|-
!Cents
|271.0
|'''309.7'''
|348.4
|'''387.1'''
|425.8
|-
!Just interpretation
|7/6 (+4.1{{c}})
|'''6/5''' (-6.0{{c}})
|11/9 (+1.0{{c}})
|'''5/4''' (+0.8{{c}})
|9/7 (-9.2{{c}})
|-
!Steps
|7
|'''8'''
|9
|'''10'''
|11
|}
Diatonic thirds are bolded.

If one wants to stretch the octave to improve JI approximation, the [https://en.xen.wiki/w/Zeta_peak_index zeta peak] octave stretch and the CWE optimization of 7-limit 31tet give an octave of 1200.8 cents, while the CWE optimization of 11-limit 31tet gives an octave of 1201.2 cents.

=== Chords ===
Along with its diatonic major and minor chords which approximate 5-limit harmony, 31edo also has a narrow but functional supermajor triad, and a well-tuned subminor triad. It also supports arto and tendo chords, with its slendric chords of [0 6 18] and [0 12 18], and has a neutral triad [0 9 18] which represents both artoneutral and tendoneutral triads in the 11- and 13-limit.

=== Regular temperaments ===
Besides [[Meantone]] (for which it provides an excellent tuning and which is shared with 19edo), 31edo also supports variations of [[Rastmic]] temperament (like 24edo), [[Slendric]] (like 36edo), [[Miracle]] (like 41edo), and [[Orwell]] (like 22edo).

== Scales ==
31edo does not temper out 64/63, meaning that it can be used to tune [[Diasem]] while representing some simpler 5-limit intervals. 31edo's step is called a [[diesis]], and can function as an [[aberrisma]]. Due to being a prime number, every MOS scale in 31edo has a full-octave period, and 31edo has a large number of them. Orwell[9] ([[gramitonic]]) is one example, so is Mohajira[7] ([[mosh]]).

31edo also has a usable 12-note chromatic scale, approximating [[Golden sequences and tuning|golden]] Meantone/monocot.

=== MOS scales ===
A key aspect of 31edo noted by several sources is the diversity of MOS scales represented. The 15 edo-distinct regular temperaments of 31edo are divided into three loops that are traversed by halving or doubling their generators.
{| class="wikitable"
|+
!
! colspan="5" |Loop 1
|-
!Temperament
|[[Didacus]]
|[[Wurschmidt]]
|[[Squares]]
|[[Mohajira]]
|[[Meantone]]
|-
!Complexity of 3/2
|15
|8
|4
|2
|1
|-
!Scale (albitonic)
|5-5-5-5-5-6
|8-1-1-8-1-1-8-1-1-1
|2-2-7-2-2-7-2-7
|5-4-5-4-5-4-4
|5-5-3-5-5-5-3
|-
!Generator
|5, 26
|10, 21
|20, 11
|22, 9
|18, 13
|-
!
! colspan="5" |Loop 2
|-
!Temperament
|[[Miracle]]
|[[Slendric]]
|[[A-Team]]
|[[Orwell]]
|[[Casablanca]]
|-
!Complexity of 3/2
|6
|3
|14
|7
|12
|-
!Scale (albitonic)
|3-3-3-3-3-3-3-3-3-4
|6-6-6-6-7
|2-5-2-5-5-2-5-5
|4-3-4-3-4-3-4-3-3
|3-3-3-3-5-3-3-3-5
|-
!Generator
|3, 28
|6, 25
|12, 19
|24, 7
|17, 14
|-
!
! colspan="5" |Loop 3
|-
!Temperament
|[[Slender]]
|[[Carlos Alpha|Valentine]]
|[[Nusecond]]
|[[Myna]]
|[[Tritonic]]
|-
!Complexity of 3/2
|13
|9
|11
|10
|5
|-
!Scale (albitonic)
| - (11-note scale has >10 steps interval)
| - (11-note scale has >10 steps interval)
|4-4-4-4-4-4-4-3
|1-1-6-1-1-6-1-1-6-1-6
| - (11-note scale has >10 steps interval)
|-
!Generator
|1, 30
|2, 29
|4, 27
|8, 23
|15, 16
|}

== Notation ==
31edo, as one of the more popular edos, has a somewhat agreed-upon notation system. This notation is simply neutral [[diatonic notation]] applied to the edo, where a half-# or half-b represents an alteration by one diesis. In this manner, all notes can be spelled in a way that does not require multiple sharps or flats.

{| class="wikitable"
|+
!Step
!Cents
!ADIN*
!Neutral diatonic
!Notation
!Just intervals represented
|-
|0
|0.00
|unison
|unison
|A
|1/1
|-
|1
|38.71
|superunison
|semiaugmented unison
|At
|49/48, 50/49, 128/125
|-
|2
|77.42
|subminor second
|semidiminished second
|A#
|25/24
|-
|3
|116.13
|nearminor second
|minor second
|Bb
|16/15
|-
|4
|154.84
|neutral second
|neutral second
|Bd
|11/10, 12/11
|-
|5
|193.55
|nearmajor second
|major second
|B
|10/9, 9/8
|-
|6
|232.26
|supermajor second
|semiaugmented second
|Bt
|8/7
|-
|7
|270.97
|subminor third
|semidiminished third
|Cd
|7/6
|-
|8
|309.68
|nearminor third
|minor third
|C
|6/5
|-
|9
|348.39
|neutral third
|neutral third
|Ct
|11/9, 16/13
|-
|10
|387.10
|nearmajor third
|major third
|C#
|5/4
|-
|11
|425.81
|supermajor third
|semiaugmented third
|Db
|9/7
|-
|12
|464.52
|subfourth
|semidiminished fourth
|Dd
|21/16, 13/10
|-
|13
|503.23
|perfect fourth
|perfect fourth
|D
|4/3
|-
|14
|541.94
|neutral fourth
|semiaugmented fourth
|Dt
|11/8, 15/11
|-
|15
|580.65
|nearaugmented fourth
|augmented fourth
|D#
|7/5
|-
|16
|619.35
|neardiminished fifth
|diminished fifth
|Eb
|10/7
|-
|17
|658.06
|neutral fifth
|semidiminished fifth
|Ed
|16/11, 22/15
|-
|18
|696.77
|perfect fifth
|perfect fifth
|E
|3/2
|-
|19
|735.48
|superfifth
|semiaugmented fifth
|Et
|32/21, 20/13
|-
|20
|774.19
|subminor sixth
|semidiminished sixth
|Fd
|14/9
|-
|21
|812.90
|nearminor sixth
|minor sixth
|F
|8/5
|-
|22
|851.61
|neutral sixth
|neutral sixth
|Ft
|13/8, 18/11
|-
|23
|890.32
|nearmajor sixth
|major sixth
|F#
|5/3
|-
|24
|929.03
|supermajor sixth
|semiaugmented sixth
|Gb
|12/7
|-
|25
|967.74
|subminor seventh
|semidiminished seventh
|Gd
|7/4
|-
|26
|1006.45
|nearminor seventh
|minor seventh
|G
|9/5, 16/9
|-
|27
|1045.16
|neutral seventh
|neutral seventh
|Gt
|11/6, 20/11
|-
|28
|1083.87
|nearmajor seventh
|major seventh
|G#
|15/8
|-
|29
|1122.58
|supermajor seventh
|semiaugmented seventh
|Ab
|48/25
|-
|30
|1161.29
|suboctave
|semidiminished octave
|Ad
|49/25, 125/64, 96/49
|-
|31
|1200.00
|octave
|octave
|A
|2/1
|}
<nowiki>*</nowiki>Aligned with the consensus agreed upon by various 31edo resources. ADIN in specific clarifies the unspecified "major" and "minor" qualities found in these resources as "nearmajor" and "nearminor" to distinguish them from supermajor/subminor.

== SCL files ==
See [[31edo/SCL files]].

== Multiples ==
=== 217edo ===
217edo is a theoretically strong system which keeps 31edo's tuning of 2.5.7.(13:17:19). 217edo is strong in the 19-limit and the smallest edo distinctly consistent in the 19-odd-limit.

{{Harmonics in ED|217|37|0}}
{{Cat|Edos}}{{Navbox EDO}}

31edo

2026-05-20T19:07:09Z

Hkm:

[[File:31edo whr.png|thumb|418x418px|31edo supports Valentine and Miracle, alongside supporting Meantone.]]
'''31edo''', or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 39 [[cent]]s. It is most commonly known as a tuning of [[Meantone]], and for its accurate approximation of the 2.5.7 [[subgroup]].

31edo as a whole contains a diverse palette of interval qualities and structures ranging from the very familiar to the quite exotic, and remarkably, almost all of these still have a reasonably simple harmonic interpretation. As a meantone system, 31edo's diatonic scale includes the basic qualities of the [[5-limit]], such as the [[perfect fourth]] and [[perfect fifth|fifth]], and the classical minor and major thirds ([[6/5]] and [[5/4]]). But 31edo also includes subminor and supermajor intervals, identifiable with [[septal]] ratios such as [[7/6]] and [[9/7]], and [[neutral]] intervals, identifiable with [[11-limit]] ratios such as [[11/9]].

In terms of structures, or ways of organizing harmony, it should first be noted that 31edo's perfect fifth, of 18 steps, is quite divisible. The fifth can be split in two, giving us a neutral-third temperament, known in this case as [[Mohajira]], which emphasizes heptatonic structure, the 11th harmonic, and 2.3.5.11. Splitting the fifth in three gives us [[Slendric]] (or in this case [[Mothra]]), formed by stacking [[8/7]], and which emphasizes pentatonic structure and 2.3.7. Combining these gives us [[Miracle]], while splitting Slendric into three again gives us [[Valentine]]. In 31edo, each of these provides xenharmonic ways of accessing the 11-limit with more simplicity than Meantone. Yet another way to encompass the 11-limit is given by [[Orwell]], generated by 31edo's subminor third; and of course, as a prime EDO, 31edo contains several more structures unique to itself.

== Theory ==
==== Edostep interpretations ====
31edo's edostep has the following interpretations in the 11-limit:

* 128/125 (the residue between three stacked 5/4s and the octave)
* 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)
* 50/49 (the difference between [[7/5]] and [[10/7]])
* 64/63 (the difference between 8/7 and [[9/8]])
* 33/32 (the difference between [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or [[11/10]] and 9/8)
* 55/54 (the difference between 6/5 and 11/9, or 12/11 and [[10/9]])
* 56/55 (the difference between 5/4 and [[14/11]])

==== JI approximation ====
31edo can be understood as a 7-limit system with a somewhat flat 3/2 but nearly-perfect 5th and 7th harmonics. In particular, the product of 5 and 7, [[35/32]], is approximated to within about 0.3{{c}}. 31edo also has an approximation to the 11th harmonic that, while tuned flat, has the property that the flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral third. The harmonic 23 turns out to be flat in a very similar way to 11.

The intervening harmonics - 13, 17, and 19 - are tuned rather sharp, but by almost exactly the same amount; therefore the chord 13:17:19 is extremely well-approximated by 31edo, with the interval [[17/13]] tuned less than 0.1{{c}} off.

31edo's fifth generates a functional diatonic scale. Its whole tone, of 5 steps, is split into semitones of 2 and 3; as 31edo's fifth is flatter than that of [[12edo]], the chromatic semitone, comprised by 2 steps, is smaller than the diatonic semitone, which is 3 steps.
{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality ([[ADIN]])
|Subminor
|'''Nearminor'''
|Neutral
|'''Nearmajor'''
|Supermajor
|-
!Cents
|271.0
|'''309.7'''
|348.4
|'''387.1'''
|425.8
|-
!Just interpretation
|7/6 (+4.1{{c}})
|'''6/5''' (-6.0{{c}})
|11/9 (+1.0{{c}})
|'''5/4''' (+0.8{{c}})
|9/7 (-9.2{{c}})
|-
!Steps
|7
|'''8'''
|9
|'''10'''
|11
|}
Diatonic thirds are bolded.

If one wants to stretch the octave to improve all-around JI approximation, the [https://en.xen.wiki/w/Zeta_peak_index zeta peak] octave stretch and the CWE optimization of 7-limit 31tet give an octave of 1200.8 cents, while the CWE optimization of 11-limit 31tet gives an octave of 1201.2 cents.

=== Chords ===
Along with its diatonic major and minor chords which approximate 5-limit harmony, 31edo also has a narrow but functional supermajor triad, and a well-tuned subminor triad. It also supports arto and tendo chords, with its slendric chords of [0 6 18] and [0 12 18], and has a neutral triad [0 9 18] which represents both artoneutral and tendoneutral triads in the 11- and 13-limit.

=== Regular temperaments ===
Besides [[Meantone]] (for which it provides an excellent tuning and which is shared with 19edo), 31edo also supports variations of [[Rastmic]] temperament (like 24edo), [[Slendric]] (like 36edo), [[Miracle]] (like 41edo), and [[Orwell]] (like 22edo).

== Scales ==
31edo does not temper out 64/63, meaning that it can be used to tune [[Diasem]] while representing some simpler 5-limit intervals. 31edo's step is called a [[diesis]], and can function as an [[aberrisma]]. Due to being a prime number, every MOS scale in 31edo has a full-octave period, and 31edo has a large number of them. Orwell[9] ([[gramitonic]]) is one example, so is Mohajira[7] ([[mosh]]).

31edo also has a usable 12-note chromatic scale, approximating [[Golden sequences and tuning|golden]] Meantone/monocot.

=== MOS scales ===
A key aspect of 31edo noted by several sources is the diversity of MOS scales represented. The 15 edo-distinct regular temperaments of 31edo are divided into three loops that are traversed by halving or doubling their generators.
{| class="wikitable"
|+
!
! colspan="5" |Loop 1
|-
!Temperament
|[[Didacus]]
|[[Wurschmidt]]
|[[Squares]]
|[[Mohajira]]
|[[Meantone]]
|-
!Complexity of 3/2
|15
|8
|4
|2
|1
|-
!Scale (albitonic)
|5-5-5-5-5-6
|8-1-1-8-1-1-8-1-1-1
|2-2-7-2-2-7-2-7
|5-4-5-4-5-4-4
|5-5-3-5-5-5-3
|-
!Generator
|5, 26
|10, 21
|20, 11
|22, 9
|18, 13
|-
!
! colspan="5" |Loop 2
|-
!Temperament
|[[Miracle]]
|[[Slendric]]
|[[A-Team]]
|[[Orwell]]
|[[Casablanca]]
|-
!Complexity of 3/2
|6
|3
|14
|7
|12
|-
!Scale (albitonic)
|3-3-3-3-3-3-3-3-3-4
|6-6-6-6-7
|2-5-2-5-5-2-5-5
|4-3-4-3-4-3-4-3-3
|3-3-3-3-5-3-3-3-5
|-
!Generator
|3, 28
|6, 25
|12, 19
|24, 7
|17, 14
|-
!
! colspan="5" |Loop 3
|-
!Temperament
|[[Slender]]
|[[Carlos Alpha|Valentine]]
|[[Nusecond]]
|[[Myna]]
|[[Tritonic]]
|-
!Complexity of 3/2
|13
|9
|11
|10
|5
|-
!Scale (albitonic)
| - (11-note scale has >10 steps interval)
| - (11-note scale has >10 steps interval)
|4-4-4-4-4-4-4-3
|1-1-6-1-1-6-1-1-6-1-6
| - (11-note scale has >10 steps interval)
|-
!Generator
|1, 30
|2, 29
|4, 27
|8, 23
|15, 16
|}

== Notation ==
31edo, as one of the more popular edos, has a somewhat agreed-upon notation system. This notation is simply neutral [[diatonic notation]] applied to the edo, where a half-# or half-b represents an alteration by one diesis. In this manner, all notes can be spelled in a way that does not require multiple sharps or flats.

{| class="wikitable"
|+
!Step
!Cents
!ADIN*
!Neutral diatonic
!Notation
!Just intervals represented
|-
|0
|0.00
|unison
|unison
|A
|1/1
|-
|1
|38.71
|superunison
|semiaugmented unison
|At
|49/48, 50/49, 128/125
|-
|2
|77.42
|subminor second
|semidiminished second
|A#
|25/24
|-
|3
|116.13
|nearminor second
|minor second
|Bb
|16/15
|-
|4
|154.84
|neutral second
|neutral second
|Bd
|11/10, 12/11
|-
|5
|193.55
|nearmajor second
|major second
|B
|10/9, 9/8
|-
|6
|232.26
|supermajor second
|semiaugmented second
|Bt
|8/7
|-
|7
|270.97
|subminor third
|semidiminished third
|Cd
|7/6
|-
|8
|309.68
|nearminor third
|minor third
|C
|6/5
|-
|9
|348.39
|neutral third
|neutral third
|Ct
|11/9, 16/13
|-
|10
|387.10
|nearmajor third
|major third
|C#
|5/4
|-
|11
|425.81
|supermajor third
|semiaugmented third
|Db
|9/7
|-
|12
|464.52
|subfourth
|semidiminished fourth
|Dd
|21/16, 13/10
|-
|13
|503.23
|perfect fourth
|perfect fourth
|D
|4/3
|-
|14
|541.94
|neutral fourth
|semiaugmented fourth
|Dt
|11/8, 15/11
|-
|15
|580.65
|nearaugmented fourth
|augmented fourth
|D#
|7/5
|-
|16
|619.35
|neardiminished fifth
|diminished fifth
|Eb
|10/7
|-
|17
|658.06
|neutral fifth
|semidiminished fifth
|Ed
|16/11, 22/15
|-
|18
|696.77
|perfect fifth
|perfect fifth
|E
|3/2
|-
|19
|735.48
|superfifth
|semiaugmented fifth
|Et
|32/21, 20/13
|-
|20
|774.19
|subminor sixth
|semidiminished sixth
|Fd
|14/9
|-
|21
|812.90
|nearminor sixth
|minor sixth
|F
|8/5
|-
|22
|851.61
|neutral sixth
|neutral sixth
|Ft
|13/8, 18/11
|-
|23
|890.32
|nearmajor sixth
|major sixth
|F#
|5/3
|-
|24
|929.03
|supermajor sixth
|semiaugmented sixth
|Gb
|12/7
|-
|25
|967.74
|subminor seventh
|semidiminished seventh
|Gd
|7/4
|-
|26
|1006.45
|nearminor seventh
|minor seventh
|G
|9/5, 16/9
|-
|27
|1045.16
|neutral seventh
|neutral seventh
|Gt
|11/6, 20/11
|-
|28
|1083.87
|nearmajor seventh
|major seventh
|G#
|15/8
|-
|29
|1122.58
|supermajor seventh
|semiaugmented seventh
|Ab
|48/25
|-
|30
|1161.29
|suboctave
|semidiminished octave
|Ad
|49/25, 125/64, 96/49
|-
|31
|1200.00
|octave
|octave
|A
|2/1
|}
<nowiki>*</nowiki>Aligned with the consensus agreed upon by various 31edo resources. ADIN in specific clarifies the unspecified "major" and "minor" qualities found in these resources as "nearmajor" and "nearminor" to distinguish them from supermajor/subminor.

== SCL files ==
See [[31edo/SCL files]].

== Multiples ==
=== 217edo ===
217edo is a theoretically strong system which keeps 31edo's tuning of 2.5.7.(13:17:19). 217edo is strong in the 19-limit and the smallest edo distinctly consistent in the 19-odd-limit.

{{Harmonics in ED|217|37|0}}
{{Cat|Edos}}{{Navbox EDO}}

User:Hkm/common.css

2026-05-18T02:44:33Z

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.thl, .wikitable .thl {
color: #000 !important;
background-color: #cba !important;
}
.prime2 {
background-color: #BFBFBF !important;
color: black !important;
}
.prime3 {
background-color: #F17F83 !important;
color: black !important;
}
.prime5 {
background-color: #D0FEAA !important;
color: black !important;
}
.prime7 {
background-color: #987FD1 !important;
color: black !important;
}
.prime11 {
background-color: #FFE48E !important;
color: black !important;
}
.prime13 {
background-color: #DC7FDD !important;
color: black !important;
}
.prime17 {
background-color: #87CEEA !important;
color: black !important;
}
.prime19 {
background-color: #9DEAB4 !important;
color: black !important;
}
.prime23 {
background-color: #E9AE74 !important;
color: black !important;
}
.prime29 {
background-color: #7C72C0 !important;
color: black !important;
}
.prime31 {
background-color: #729FCA !important;
color: black !important;
}
.prime37 {
background-color: #A0E0D0 !important;
color: black !important;
}
.prime41 {
background-color: #D0E099 !important;
color: black !important;
}
.prime43 {
background-color: #E0D391 !important;
color: black !important;
}
.prime47 {
background-color: #DF987C !important;
color: black !important;
}
.prime53 {
background-color: #B17CC5 !important;
color: black !important;

Temperament

2026-03-23T03:34:06Z

Hkm:

'''Temperament''' is a method of tuning musical instruments based on approximating harmonic targets (almost always [[Just intonation|just intervals]]) with other intervals, in order to maintain the desired harmony between sounds while simultaneously simplifying melodic structure. For example, in standard tuning, we approximate the harmonic target of 3/2 with the 7-semitone interval, and the target of 5/4 with the 4-semitone interval. The process of application of temperament is called '''tempering'''.

Temperaments can be defined as sequences of operations or more formally as mathematical functions. Most commonly-used temperaments can be divided into [[Irregular temperament|well temperaments]] and [[regular temperament]]s. [[Equal temperament|Equal temperaments]], a very popular tuning approach, are a subset of regular temperaments.

== History ==
In Medieval Western Europe, [[Pythagorean tuning]] was the most widely used, in which fifths and fourths are tuned to just intonation. At the time, thirds were omitted and considered a dissonance, which followed from the characteristics of Pythagorean tuning (the [[diatonic major third]] is rather complex). Instrumental music was based on two-part voice leading. To overcome these limitations, a compromise solution was sought that would allow the use of just thirds without sacrificing the purity of fifths.

That lead to the adoption of [[Meantone|meantone temperament]] in the end of the 15th century. Fifths began to be narrowed, which led to the complete disappearance of the just fifth, and instead the impression of purity of harmony was achieved by just thirds. This enabled the introduction of third-based chords and development of [[tertian harmony]].

The downside of meantone was that not every diatonic scale known in European music in that time could be played in tune on tempered instruments. In practice, meantone was being modified in various ways by tuners to reduce the dissonance created by [[Wolf interval|wolf intervals]]. Some theoreticians since the 16th century proposed to resolve the problem by the use of equal temperaments like [[12edo]], [[19edo]], and [[31edo]].

12-tone equal temperament became widely accepted practice by the beginning of the 18th century.

[[Category:Core knowledge]]

Semifourth-generated scales

2026-03-22T18:52:42Z

Hkm:

{{problematic}}

'''Semifourth-generated scales''' are scales in tuning systems where two generators of approximately 250 cents in size stack to a perfect fourth, defining a temperament archetype called '''alpha-dicot''' or '''omega-dicot'''{{Adv|, and notated [P8, P4/2] in pergen notation.}}

The semifourth-generated scales are characteristically 4L 1s, semiquartal (5L4s), and 5L9s. 4L 1s is structurally significant because it is the [[Equipentatonic|equal trichordal pentatonic]].

== Intervals and notation ==
Alpha-dicot notation is complicated as it conventionally requires either the introduction of new "hemi-Pythagorean" ordinals or the use of scales other than the standard diatonic scale. As such, there is no universally accepted convention. An intuitive option, if one that takes getting used to, is to use [https://en.xen.wiki/w/KISS_notation KISS] notation or [https://en.xen.wiki/w/Diamond-mos_notation diamond-mos] notation for 5L 4s.

== See also ==
* [[Chthonic harmony]]

18edo

2026-03-22T18:43:53Z

Hkm: /* Scales */

'''18edo''', or 18 equal divisions of the octave, is the equal tuning featuring steps of (1200/18) ~= 66.7 cents, 18 of which stack to the octave 2/1.

With the sharp fifth 733.3c and the flat fifth 666.7c almost equally detuned from the just fifth, 18edo is often considered the quintessential [[straddle primes|straddle-3]] edo and the straddle-3 version of [[12edo]]. It does not approximate low harmonics well, except 9 and debatably 5; it is also straddle-7, 13, 17, and 19.

== Scales ==
In this section, notations like "2222122221" for taric refer to how many steps of 18edo are between successive degrees of the scale. For example, the first step of taric (between the first two degrees) is two steps of 18edo. The notations in parentheses, like "8L2s", are [[MOS]] or [[MOS substitution]] notation that describes the same thing as the other notation more concisely. All "view" links go to Scale Workshop, a website that lets the user play in a scale through the computer keyboard or touch-screen interface, and download .scl files for use in synths.
* Straddle-3 diatonic (5L1m1s), 3331332 or 3332331, constructed by an alternating stack of flat and sharp fifths
* [[Oneirotonic]] (5L3s), 33133131 (compressed 17edo diatonic). ([https://scaleworkshop.plainsound.org/scale/RBx7WVRn- view])
* [[Smitonic]] (4L3s), 3323232 (stretched 19edo diatonic). ([https://scaleworkshop.plainsound.org/scale/RBxvWZhiE view])
* [[Taric]] (8L2s), 2222122221 and the altered MOS pentachordal taric, 2221222221
* Hexawood (6L6s), 212121212121 is a "straddle-3 chromatic scale", constructed by an alternating stack of flat and sharp fifths
* Smitonic MODMOS, 2332233 (inverted step pattern of the scale associated with rast maqam). ([https://scaleworkshop.plainsound.org/scale/RBxV9E9cC view])
* Antidiatonic, 2242224 (inverted step pattern of diatonic). ([https://scaleworkshop.plainsound.org/scale/RBxzvsm6S view])

== Edostep interpretations ==

==== Edostep interpretations ====
18edo's edostep has the following interpretations in its [[patent val]]:

* 25/24 (the difference between 5/4 and 6/5)
* 56/55 (the difference between 11/8 and 7/5)
* 80/77 (the difference between 7/5 and 16/11)
* 36/35 (the difference between 6/5 and 7/6)

== JI approximation ==
18edo is straddle-3 and -7, but inherits 12edo's approximation of 5 and also approximates 11 to a similar degree of accuracy, making 55/32 tuned accurately. Additionally, 7 and 3 are both sharp about 30 cents, meaning 7/6 is tuned well. Other intervals tuned well include 28/25 and 25/24. Additionally, because 18edo has both the 5 and 7 from 12edo, 7/5 and 10/7 are tempered together ([[jubilic]] temperament) and therefore tuned to 600c (the perfect semioctave). This means that passable tunings of 5:6:7 and its utonal counterpart are available.

Additionally, 18edo's approximate 4:5:6 using the sharp fifth is approximately delta-rational (similar to the case with [[15edo]]), albeit slightly less accurate. It is roughly the isoharmonic JI chord 19:24:29. {{Harmonics in ED|18|31|0}}

== Scales and chords ==
18edo's best fifth is 733.3 cents, which generates an oneirotonic scale; it can also be used as the double of 9edo, which has an antidiatonic scale.

In oneirotonic, the "thirds" are the same degree as the "major second" and "perfect fourth" respectively, so that any oneirotonic scale always has two distinct "thirds". This follows from the generator being an odd number of steps. In 13edo, the four possible "thirds" collapse to 3 (as the "diminished fourth" and "major third" are identical) but in 18edo (and in 21edo) there are four distinct qualities, here named with [[ADIN|oneirotonic ADIN]]:

* [0 4 11] - "inframinor" (267c)
* [0 5 11] - "nearminor" (333c)
* [0 6 11] - "nearmajor" (400c)
* [0 7 11] - "ultramajor" (467c)

([https://dxinteractive.github.io/xenpaper/#%7B18edo%7D%0A(osc%3Afmtriangle)%0A%5B0_4_11%5D----_._._%5B0_5_11%5D----_._._%5B0_6_11%5D----_._._%5B0_7_11%5D----_._. listen])

The brightest mode of oneirotonic (other than the very brightest, which doesn't have the 733c fifth) has ultramajor and nearmajor. Most modes have ultramajor and inframinor, corresponding to diatonic sus4 and sus2 chords. The darkest mode has nearminor and inframinor.

Alternately, in antidiatonic, 18edo adds to 9edo a "neutral" third, which is the same as the nearminor oneirotonic third. Therefore, there are 5 different qualities of antidiatonic fifth-bounded triad:

* [0 3 10] - "inframinor" (200c)
* [0 4 10] - "farminor" (267c)
* [0 5 10] - "neutral" (333c)
* [0 6 10] - "farmajor" (400c)
* [0 7 10] - "ultramajor" (467c)

([https://dxinteractive.github.io/xenpaper/#%7B18edo%7D%0A(osc%3Afmtriangle)%0A%5B0_3_10%5D--_.._%5B0_4_10%5D--_.._%5B0_5_10%5D--_.._%5B0_6_10%5D--_.._%5B0_7_10%5D--_.._ listen])

These chords also exist in oneirotonic (except for the neutral one), as the antidiatonic fifth is also the oneirotonic major tritone, which appears on 4 different scale degrees.

Additionally, there is the dual-fifth interpretation of 18edo, wherein the basic scale is similar to a 19edo diatonic but with one edostep removed. It is useful to organize the chords based on their appearance on each degree of dual-fifth diatonic.
{| class="wikitable"
|+
!Degree
!LLsLLLm
!LLmLLLs
|-
|1
|[0 6 10] - major antidiatonic
|[0 6 11] - nearmajor
|-
|2
|[0 4 10] - minor antidiatonic
|[0 5 11] - nearminor
|-
|3
|[0 4 10] - minor antidiatonic
|[0 5 11] - nearminor
|-
|4
|[0 6 11] - nearmajor
|[0 6 10] - major antidiatonic
|-
|5
|[0 6 11] - nearmajor
|[0 6 10] - major antidiatonic
|-
|6
|[0 5 11] - nearminor
|[0 4 10] - minor antidiatonic
|-
|7
|[0 5 9] - otonal diminished
|[0 4 9] - utonal diminished
|}
The otonal and utonal diminished triads provide tunings of 5:6:7 and 1/(5:6:7) respectively, despite the fact that 18edo tunes neither 5 nor 7 particularly well (in fact, it has the same mappings as 12edo).

To use a scale which includes neutral harmony, an option would be altering some of the antidiatonic degrees to be neutral. This generates "smitonic" (3-3-2-3-2-3-2) or a MODMOS thereof, which can be thought of as the antidiatonic counterpart of [[mosh]]. The MOS form of smitonic has five of the seven tertian triads neutral; the remaining two are nearmajor and nearminor triads using the oneirotonic fifth. The MODMOS 2-3-3-2-2-3-3 introduces more variation, adding in a major and a minor antidiatonic triad, leaving neutral triads on three of the degrees.

== Delta-rational theory ==
18edo has the following approximate [[DR]] chords (below 10c pairwise logarithmic least-squares error, bounding interval < 1200c, no 1\18 or 17\18):
=== +1+1 ===
* [0 8 14]\18 (error 6.3c)
* [0 6 11]\18 (error 7.3c) - this is the "nearmajor" oneirotonic chord, which behaves like a stretched 4:5:6 in a delta-rational context (a similar chord is [0 5 9] of [[15edo]]).
=== +1+2 ===
* [0 6 15]\18 (error 1.3c)
* [0 3 8]\18 (error 4.8c)
* [0 5 13]\18 (error 8.6c)
=== +2+1 ===
* [0 7 10]\18 (error 6.8c)
* [0 11 15]\18 (error 9.5c)
=== +1+?+1 ===
* [0 4 8 11]\18 (error 0.2c)
* [0 3 11 13]\18 (error 0.2c)
* [0 7 10 15]\18 (error 3.3c)
* [0 6 12 16]\18 (error 5.2c)
* [0 6 11 15]\18 (error 6.8c) (stretched 4:5:6:7)
* [0 5 7 11]\18 (error 7.8c)
* [0 4 7 10]\18 (error 8.6c)
* [0 4 9 12]\18 (error 8.8c)

== Supersets ==
18edo's primes are mostly off by about a twelfth-tone or a sixth-tone. This implies that multiplying it by four to yield [[72edo]] yields an accurate tuning of just intonation. [[36edo]] contains 72edo's 2.3.7 subgroup and is the next level of structural resolution for said subgroup after [[5edo]]. {{Navbox EDO}}
{{Cat|Edos}}

Note entry

2026-03-22T18:31:38Z

Hkm:

'''Note entry''' refers to the ways that notes are entered into a digital audio workstation (DAW) when composing. The two most common methods are '''piano roll''' and '''notation'''.

== Piano roll ==
Many virtual instruments (for example, Vital and Dexed) support alternative tuning systems through uploading .scl files to the plugin. Once the .scl file is imported, 128 notes of the new scale are available through the piano roll in the same way that one would write notes in 12edo.

The piano roll can be utilized in several different ways depending on the preferences of the user. It is common for composers to have a maximum number of notes equal to a useful edo, such as 31, 41, 46, or 53. Because there are only 128 MIDI notes, range is a concern for larger edos.
{| class="wikitable"
!Range (Octaves)
!Max Edo
|-
|1
|128
|-
|2
|64
|-
|3
|42
|-
|4
|32
|-
|5
|25
|-
|6
|21
|-
|7
|18
|-
|8
|16
|}
Note that N octaves of range means that each octave-equivalent pitch class occurs exactly N times. The table stops at 8 octaves because that covers the entire usable pitch range, from a bass guitar with extremely heavy strings to a professional glockenspiel.

It is also possible to use midi channels to increase range. This can be done by either assigning them to octave offsets, steps of an edo superset, or a combination of both. The former is available in Surge synth by default, but the others tend to require custom scripts like the in-progress [https://spoogly.website/tools.html TuneLoon].

== Notation ==
MuseScore has a few plugins for microtuning. The one most commonly used is [https://github.com/AzureDevs/XenKit XenKit] (which supports MuseScore 3 and 4), which allows the user to specify an EDO or JI and redefines MuseScore’s built-in accidentals to mean alterations within that tuning system according to standard notation. MuseScore by default does not allow multiple accidentals to be placed on the same note (as would be necessary for JI notation), but XenKit allows these extra accidentals to be applied by writing them as lyrics.
{{Cat|
Core knowledge
Praxis
}}

Note entry

2026-03-22T18:31:26Z

Hkm: /* Piano roll */

{{problematic}}

'''Note entry''' refers to the ways that notes are entered into a digital audio workstation (DAW) when composing. The two most common methods are '''piano roll''' and '''notation'''.

== Piano roll ==
Many virtual instruments (for example, Vital and Dexed) support alternative tuning systems through uploading .scl files to the plugin. Once the .scl file is imported, 128 notes of the new scale are available through the piano roll in the same way that one would write notes in 12edo.

The piano roll can be utilized in several different ways depending on the preferences of the user. It is common for composers to have a maximum number of notes equal to a useful edo, such as 31, 41, 46, or 53. Because there are only 128 MIDI notes, range is a concern for larger edos.
{| class="wikitable"
!Range (Octaves)
!Max Edo
|-
|1
|128
|-
|2
|64
|-
|3
|42
|-
|4
|32
|-
|5
|25
|-
|6
|21
|-
|7
|18
|-
|8
|16
|}
Note that N octaves of range means that each octave-equivalent pitch class occurs exactly N times. The table stops at 8 octaves because that covers the entire usable pitch range, from a bass guitar with extremely heavy strings to a professional glockenspiel.

It is also possible to use midi channels to increase range. This can be done by either assigning them to octave offsets, steps of an edo superset, or a combination of both. The former is available in Surge synth by default, but the others tend to require custom scripts like the in-progress [https://spoogly.website/tools.html TuneLoon].

== Notation ==
MuseScore has a few plugins for microtuning. The one most commonly used is [https://github.com/AzureDevs/XenKit XenKit] (which supports MuseScore 3 and 4), which allows the user to specify an EDO or JI and redefines MuseScore’s built-in accidentals to mean alterations within that tuning system according to standard notation. MuseScore by default does not allow multiple accidentals to be placed on the same note (as would be necessary for JI notation), but XenKit allows these extra accidentals to be applied by writing them as lyrics.
{{Cat|
Core knowledge
Praxis
}}

L'Antica Musica

2026-03-22T05:39:08Z

Hkm: /* On Practice V: on the Archicembalo Instrument */

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament.
== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[Just Intonation|just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music.

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct reason by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Even Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

=== Enharmonic modes ===
{{Wip}}

== On Practice V: on the Archicembalo Instrument ==
Because ''On Practice IV'' is spend describing notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.

Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.

The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.

Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord. The fifth size was likely variable throughout. Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.

Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯.

In the second tuning, it is likely that Vicentino preferred tunings of the fifth between 1/3-comma and 2/7-comma; in the first tuning, it is likely that Vicentino preferred tunings near 1/4-comma.

Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; if the lower keyboard were tuned to 2/7-comma meantone, then ^C in this tuning would be a 2/7 of a syntonic comma higher in pitch than C. This tuning yields two wolf fifths, one between B♯ and G♭ in the lower keyboard, and one between ^B♯ and ^G♭ in the upper keyboard.

=== Interval ratios ===
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo:
{| class="wikitable"
|+Ratios of the 31-form
!Step
!Name
!Ratios (5-limit)
!Ratios (7-limit)
!Ratios (higher)
|-
|0
|Comma
|81/80
|126/125
|105/104, 385/384
|-
|1
|Diesis
|128/125, 648/625
|36/35, 50/49, 64/63
|40/39
|-
|2
|Chromatic Semitone
|25/24
|21/20
|65/63
|-
|3
|Diatonic Semitone
|16/15
|15/14
|14/13
|-
|4
|Minor Tone
| colspan="2" |N/A
|13/12
|-
|5
|Natural Tone
| colspan="3" |9/8, 10/9
|-
|6
|Major Tone
|N/A
| colspan="2" |8/7
|-
|7
|Minimal Third
| colspan="3" |Unstated
|-
|8
|Minor Third
| colspan="3" |6/5
|-
|9
|Proximate Third
| colspan="2" |N/A
|11/9
|-
|10
|Major Third
| colspan="3" |5/4
|-
|11
|Supramajor Third
|N/A
| colspan="2" |9/7
|-
|12
| colspan="4" |N/A (Vicentino skips this interval in his analysis)
|-
|13
|Perfect Fourth
| colspan="3" |4/3
|-
|14
|Superfourth
| colspan="3" |Unstated
|-
|15
|Augmented Fourth
|25/18
|7/5
|39/28
|-
|16+
|Octave Complements
| colspan="3" |Inverse ratios of their octave complements
|}

L'Antica Musica

2026-03-22T05:34:00Z

Hkm: /* On Practice V: on the Archicembalo Instrument */

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament.
== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[Just Intonation|just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music.

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct reason by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Even Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

=== Enharmonic modes ===
{{Wip}}

== On Practice V: on the Archicembalo Instrument ==
Because ''On Practice IV'' is spend describing notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.

Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.

The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.

Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord. The fifth size throughout may have been variable, but it seems like most of the fifths had sizes between those of 1/3-comma and 2/7-comma meantone. Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.

Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯, though if tuned near quarter-comma, this wolf fifth can be nearly the same size as the regular tempered fifth (hence 31edo).

Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; if the lower keyboard were tuned to 2/7-comma meantone, then ^C in this tuning would be a 2/7 of a syntonic comma higher in pitch than C. This tuning yields two wolf fifths, one between B♯ and G♭ in the lower keyboard, and one between ^B♯ and ^G♭ in the upper keyboard.

=== Interval ratios ===
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo:
{| class="wikitable"
|+Ratios of the 31-form
!Step
!Name
!Ratios (5-limit)
!Ratios (7-limit)
!Ratios (higher)
|-
|0
|Comma
|81/80
|126/125
|105/104, 385/384
|-
|1
|Diesis
|128/125, 648/625
|36/35, 50/49, 64/63
|40/39
|-
|2
|Chromatic Semitone
|25/24
|21/20
|65/63
|-
|3
|Diatonic Semitone
|16/15
|15/14
|14/13
|-
|4
|Minor Tone
| colspan="2" |N/A
|13/12
|-
|5
|Natural Tone
| colspan="3" |9/8, 10/9
|-
|6
|Major Tone
|N/A
| colspan="2" |8/7
|-
|7
|Minimal Third
| colspan="3" |Unstated
|-
|8
|Minor Third
| colspan="3" |6/5
|-
|9
|Proximate Third
| colspan="2" |N/A
|11/9
|-
|10
|Major Third
| colspan="3" |5/4
|-
|11
|Supramajor Third
|N/A
| colspan="2" |9/7
|-
|12
| colspan="4" |N/A (Vicentino skips this interval in his analysis)
|-
|13
|Perfect Fourth
| colspan="3" |4/3
|-
|14
|Superfourth
| colspan="3" |Unstated
|-
|15
|Augmented Fourth
|25/18
|7/5
|39/28
|-
|16+
|Octave Complements
| colspan="3" |Inverse ratios of their octave complements
|}

EDO

2026-03-22T05:25:33Z

Hkm:

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |''none available in basic subgroup''
|2
|-
|2.<3.>>5.>>7.<17
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.<17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5
|-
|2.9.5.>7
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.7.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.>3.<5.>>7.<11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports magic and archy with its sharp fifth and deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of schismic analogue of didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with buzzard temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-garibaldi schismic system (in fact, it supports Sensi), or a straddle-7 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|72
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|94
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|
|
|
|
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[schismic]] and [[Slendric|gamelic]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

EDO

2026-03-22T05:22:08Z

Hkm:

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |''none available in basic subgroup''
|2
|-
|2.<3.>>5.>>7.<17
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.<17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5.7.23
|-
|2.xx3.5.7.<11.>13
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.7.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.3.5.7>.11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports magic and archy with its sharp fifth and deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of schismic analogue of didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with buzzard temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-garibaldi schismic system (in fact, it supports Sensi), or a straddle-7 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|72
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|94
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|
|
|
|
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[schismic]] and [[Slendric|gamelic]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

EDO

2026-03-22T05:21:25Z

Hkm:

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |''none available in basic subgroup''
|2
|-
|2.<3.>>5.>>7.<17
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.<17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5.7.23
|-
|2.xx3.5.7.<11.>13.23
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19.23
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.7.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.3.5.7>.11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports magic and archy with its sharp fifth and deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of schismic analogue of didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with buzzard temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-garibaldi schismic system (in fact, it supports Sensi), or a straddle-7 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|72
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|94
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|
|
|
|
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[schismic]] and [[Slendric|gamelic]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

EDO

2026-03-22T05:20:42Z

Hkm:

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |''none available in basic subgroup''
|2
|-
|2.<3.>>5.>>7.<17
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.>17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5.7.23
|-
|2.xx3.5.7.<11.>13.23
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19.23
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.7.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.3.5.7>.11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports magic and archy with its sharp fifth and deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of schismic analogue of didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with buzzard temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-garibaldi schismic system (in fact, it supports Sensi), or a straddle-7 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|72
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|94
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|
|
|
|
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[schismic]] and [[Slendric|gamelic]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

EDO

2026-03-22T05:19:57Z

Hkm:

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |''none available in basic subgroup''
|2
|-
|2.<3.>>5.>>7.>17
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.>17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5.7.23
|-
|2.xx3.5.7.<11.>13.23
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19.23
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.7.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.3.5.7>.11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports magic and archy with its sharp fifth and deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of schismic analogue of didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with buzzard temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-garibaldi schismic system (in fact, it supports Sensi), or a straddle-7 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|72
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|94
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|
|
|
|
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[schismic]] and [[Slendric|gamelic]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

2.3.35 and 2.3.49

2026-03-20T00:05:13Z

Hkm: whoops that isn’t a template

{{problematic}}

'''2.3.35''' and '''2.3.49''' are two subsets of the 2.3.5.7 (septimal) group that involve many ratios further from 12edo than 2.3.7. The 35th harmonic (5*7) perfectly combines the deviations of the 5th and 7th harmonics to be about as far from 12edo as possible, whereas the 49th harmonic (7*7) overshoots slightly.

==Intervals==

===2.3.35===

2.3.35 intervals are the difference between a 5-over and 7-under interval, or vice versa.

36/35 septimal quarter tone (flat of 28/27 by 245/243)

35/32 septimal neutral second

81/70 septimal semifourth

Some of the best 2.3.5.7 scales with significant emphasis on 35 are a [[Dimension|rank-3 scale]] consisting of either [[List of regular temperaments|Meantone or Archy]] diatonic with a 35/32 offset. One small edo to do something like this is 25, which uses the notes of 5edo to approximate Archy. Stacking two offsets to get 10L5s is particularly useful.

===2.3.5.49===

2.3.49 intervals are the difference between a 7-over and 7-under interval. 49/5 is further from 12edo.

54/49 larger neutral second

49/45 smaller neutral second

49/40 neutral third

2.3.5.49 tripentatonic 5L2m5s3a (blackdye A Aeolian with added chain of 5/49 offset)

81/80

54/49

9/8

6/5

64/49

4/3

27/20

72/49

3/2

8/5

256/147

16/9

9/5

96/49

2/1

<adv>Tempering out 245/243 equates the two smallest steps, leaving 5L2m8s. Tempering out 1029/1024 equates 256/147 to 7/4. Tempering out both commas leads to Rodan temperament (2.3.5.7 41 & 46), an extension of Slendric which also notably tempers out 5120/5103.</adv>

{| class="wikitable"
|+ Over-49 Chain
|-
! 3^ !! /49 !! *245/243 !! *1029/1024
|-
| -1 || 256/147 || 1280/729 || 7/4
|-
| 0 || 64/49 || 320/243 || 21/16
|-
| 1 || 96/49 || 160/81 || 63/32
|-
| 2 || 72/49 || 40/27 || 189/128
|-
| 3 || 54/49 || 10/9 || 567/512
|-
| 4 || 81/49 || 5/3 || 1701/1024
|-
| 5 || 243/196 || 5/4 || 5103/4096
|}

{{Cat|JI groups}}

2.3.35 and 2.3.49

2026-03-20T00:04:52Z

Hkm:

{{problematic}} {{delete}}

'''2.3.35''' and '''2.3.49''' are two subsets of the 2.3.5.7 (septimal) group that involve many ratios further from 12edo than 2.3.7. The 35th harmonic (5*7) perfectly combines the deviations of the 5th and 7th harmonics to be about as far from 12edo as possible, whereas the 49th harmonic (7*7) overshoots slightly.

==Intervals==

===2.3.35===

2.3.35 intervals are the difference between a 5-over and 7-under interval, or vice versa.

36/35 septimal quarter tone (flat of 28/27 by 245/243)

35/32 septimal neutral second

81/70 septimal semifourth

Some of the best 2.3.5.7 scales with significant emphasis on 35 are a [[Dimension|rank-3 scale]] consisting of either [[List of regular temperaments|Meantone or Archy]] diatonic with a 35/32 offset. One small edo to do something like this is 25, which uses the notes of 5edo to approximate Archy. Stacking two offsets to get 10L5s is particularly useful.

===2.3.5.49===

2.3.49 intervals are the difference between a 7-over and 7-under interval. 49/5 is further from 12edo.

54/49 larger neutral second

49/45 smaller neutral second

49/40 neutral third

2.3.5.49 tripentatonic 5L2m5s3a (blackdye A Aeolian with added chain of 5/49 offset)

81/80

54/49

9/8

6/5

64/49

4/3

27/20

72/49

3/2

8/5

256/147

16/9

9/5

96/49

2/1

<adv>Tempering out 245/243 equates the two smallest steps, leaving 5L2m8s. Tempering out 1029/1024 equates 256/147 to 7/4. Tempering out both commas leads to Rodan temperament (2.3.5.7 41 & 46), an extension of Slendric which also notably tempers out 5120/5103.</adv>

{| class="wikitable"
|+ Over-49 Chain
|-
! 3^ !! /49 !! *245/243 !! *1029/1024
|-
| -1 || 256/147 || 1280/729 || 7/4
|-
| 0 || 64/49 || 320/243 || 21/16
|-
| 1 || 96/49 || 160/81 || 63/32
|-
| 2 || 72/49 || 40/27 || 189/128
|-
| 3 || 54/49 || 10/9 || 567/512
|-
| 4 || 81/49 || 5/3 || 1701/1024
|-
| 5 || 243/196 || 5/4 || 5103/4096
|}

{{Cat|JI groups}}

18edo

2026-03-19T20:02:27Z

Hkm: Undo revision 5078 by Hkm (talk)

{{problematic}}

'''18edo''', or 18 equal divisions of the octave, is the equal tuning featuring steps of (1200/18) ~= 66.7 cents, 18 of which stack to the octave 2/1.

With the sharp fifth 733.3c and the flat fifth 666.7c almost equally detuned from the just fifth, 18edo is often considered the quintessential [[straddle primes|straddle-3]] edo and the straddle-3 version of [[12edo]]. It does not approximate low harmonics well, except 9 and debatably 5; it is also straddle-7, 13, 17, and 19.

== Scales ==
* Straddle-3 diatonic (5L1m1s), 3331332 or 3332331, constructed by an alternating stack of flat and sharp fifths
* [[Oneirotonic]] (5L3s), 33133131 (compressed 17edo diatonic)
* [[Smitonic]] (4L3s), 3323232 (stretched 19edo diatonic)
* [[Taric]] (8L2s), 2222122221 and the altered MOS pentachordal taric, 2221222221
* Hexawood (6L6s) is a "straddle-3 chromatic scale", constructed by an alternating stack of flat and sharp fifths

== Edostep interpretations ==

==== Edostep interpretations ====
18edo's edostep has the following interpretations in its patent val:

* 25/24 (the difference between 5/4 and 6/5)
* 56/55 (the difference between 11/8 and 7/5)
* 80/77 (the difference between 7/5 and 16/11)
* 36/35 (the difference between 6/5 and 7/6)

== JI approximation ==
18edo is straddle-3 and -7, but inherits 12edo's approximation of 5 and also approximates 11 to a similar degree of accuracy, making 55/32 tuned accurately. Additionally, 7 and 3 are both sharp about 30 cents, meaning 7/6 is tuned well. Oher intervals tuned well include 28/25 and 25/24. Additionally, because 18edo has both the 5 and 7 from 12edo, 7/5 and 10/7 are tempered together ([[jubilic]] temperament) and therefore tuned to 600c (the perfect semioctave). This means that passable tunings of 5:6:7 and its utonal counterpart are available.

Additionally, 18edo's approximate 4:5:6 using the sharp fifth is approximately delta-rational (similar to the case with [[15edo]]), albeit slightly less accurate. It is roughly the isoharmonic JI chord 19:24:29. {{Harmonics in ED|18|31|0}}

== Non-DR theory ==
18edo's best fifth is 733.3 cents, which generates an oneirotonic scale; it can also be used as the double of 9edo, which has an antidiatonic scale.

In oneirotonic, the "thirds" are the same degree as the "major second" and "perfect fourth" respectively, so that any oneirotonic scale always has two distinct "thirds". This follows from the generator being an odd number of steps. In 13edo, the four possible "thirds" collapse to 3 (as the "diminished fourth" and "major third" are identical) but in 18edo (and in 21edo) there are four distinct qualities, here named with [[ADIN|oneirotonic ADIN]]:

* [0 4 11] - "inframinor" (267c)
* [0 5 11] - "nearminor" (333c)
* [0 6 11] - "nearmajor" (400c)
* [0 7 11] - "ultramajor" (467c)

([https://dxinteractive.github.io/xenpaper/#%7B18edo%7D%0A(osc%3Afmtriangle)%0A%5B0_4_11%5D----_._._%5B0_5_11%5D----_._._%5B0_6_11%5D----_._._%5B0_7_11%5D----_._. listen])

The brightest mode of oneirotonic (other than the very brightest, which doesn't have the 733c fifth) has ultramajor and nearmajor. Most modes have ultramajor and inframinor, corresponding to diatonic sus4 and sus2 chords. The darkest mode has nearminor and inframinor.

Alternately, in antidiatonic, 18edo adds to 9edo a "neutral" third, which is the same as the nearminor oneirotonic third. Therefore, there are 5 different qualities of antidiatonic fifth-bounded triad:

* [0 3 10] - "inframinor" (200c)
* [0 4 10] - "farminor" (267c)
* [0 5 10] - "neutral" (333c)
* [0 6 10] - "farmajor" (400c)
* [0 7 10] - "ultramajor" (467c)

([https://dxinteractive.github.io/xenpaper/#%7B18edo%7D%0A(osc%3Afmtriangle)%0A%5B0_3_10%5D--_.._%5B0_4_10%5D--_.._%5B0_5_10%5D--_.._%5B0_6_10%5D--_.._%5B0_7_10%5D--_.._ listen])

These chords also exist in oneirotonic (except for the neutral one), as the antidiatonic fifth is also the oneirotonic major tritone, which appears on 4 different scale degrees.

Additionally, there is the dual-fifth interpretation of 18edo, wherein the basic scale is similar to a 19edo diatonic but with one edostep removed. It is useful to organize the chords based on their appearance on each degree of dual-fifth diatonic.
{| class="wikitable"
|+
!Degree
!LLsLLLm
!LLmLLLs
|-
|1
|[0 6 10] - major antidiatonic
|[0 6 11] - nearmajor
|-
|2
|[0 4 10] - minor antidiatonic
|[0 5 11] - nearminor
|-
|3
|[0 4 10] - minor antidiatonic
|[0 5 11] - nearminor
|-
|4
|[0 6 11] - nearmajor
|[0 6 10] - major antidiatonic
|-
|5
|[0 6 11] - nearmajor
|[0 6 10] - major antidiatonic
|-
|6
|[0 5 11] - nearminor
|[0 4 10] - minor antidiatonic
|-
|7
|[0 5 9] - otonal diminished
|[0 4 9] - utonal diminished
|}
The otonal and utonal diminished triads provide tunings of 5:6:7 and 1/(5:6:7) respectively, despite the fact that 18edo tunes neither 5 nor 7 particularly well (in fact, it has the same mappings as 12edo).

To use a scale which includes neutral harmony, an option would be altering some of the antidiatonic degrees to be neutral. This generates "smitonic" (3-3-2-3-2-3-2) or a MODMOS thereof, which can be thought of as the antidiatonic counterpart of [[mosh]]. The MOS form of smitonic has five of the seven tertian triads neutral; the remaining two are nearmajor and nearminor triads using the oneirotonic fifth. The MODMOS 2-3-3-2-2-3-3 introduces more variation, adding in a major and a minor antidiatonic triad, leaving neutral triads on three of the degrees.

=== Scale Workshop links ===
[https://scaleworkshop.plainsound.org/scale/RBxV9E9cC Smitonic MODMOS] (inverted step pattern of the scale associated with rast maqam)

[https://scaleworkshop.plainsound.org/scale/RBxvWZhiE Smitonic] (neutralized antidiatonic)

[https://scaleworkshop.plainsound.org/scale/RBxzvsm6S Antidiatonic] (inverted step pattern of diatonic)

[https://scaleworkshop.plainsound.org/scale/RBx7WVRn- Oneirotonic]

== DR theory ==
18edo has the following approximate [[DR]] chords (below 10c pairwise logarithmic least-squares error, bounding interval < 1200c, no 1\18 or 17\18):
=== +1+1 ===
* [0 8 14]\18 (error 6.3c)
* [0 6 11]\18 (error 7.3c) - this is the "nearmajor" oneirotonic chord, which behaves like a stretched 4:5:6 in a delta-rational context (a similar chord is [0 5 9] of [[15edo]]).
=== +1+2 ===
* [0 6 15]\18 (error 1.3c)
* [0 3 8]\18 (error 4.8c)
* [0 5 13]\18 (error 8.6c)
=== +2+1 ===
* [0 7 10]\18 (error 6.8c)
* [0 11 15]\18 (error 9.5c)
=== +1+?+1 ===
* [0 4 8 11]\18 (error 0.2c)
* [0 3 11 13]\18 (error 0.2c)
* [0 7 10 15]\18 (error 3.3c)
* [0 6 12 16]\18 (error 5.2c)
* [0 6 11 15]\18 (error 6.8c) (stretched 4:5:6:7)
* [0 5 7 11]\18 (error 7.8c)
* [0 4 7 10]\18 (error 8.6c)
* [0 4 9 12]\18 (error 8.8c)

== Supersets ==
18edo's primes are mostly off by about a twelfth-tone or a sixth-tone. This implies that multiplying it by four to yield [[72edo]] yields an accurate tuning of just intonation. [[36edo]] contains 72edo's 2.3.7 subgroup and is the next level of structural resolution for said subgroup after [[5edo]]. {{Navbox EDO}}
{{Cat|Edos}}

18edo

2026-03-19T20:01:43Z

Hkm: /* JI approximation */

{{problematic}}

'''18edo''', or 18 equal divisions of the octave, is the equal tuning featuring steps of (1200/18) ~= 66.7 cents, 18 of which stack to the octave 2/1.

With the sharp fifth 733.3c and the flat fifth 666.7c almost equally detuned from the just fifth, 18edo is often considered the quintessential [[straddle primes|straddle-3]] edo and the straddle-3 version of [[12edo]]. It does not approximate low harmonics well, except 9 and debatably 5; it is also straddle-7, 13, 17, and 19.

== Scales ==
* Straddle-3 diatonic (5L1m1s), 3331332 or 3332331, constructed by an alternating stack of flat and sharp fifths
* [[Oneirotonic]] (5L3s), 33133131 (compressed 17edo diatonic)
* [[Smitonic]] (4L3s), 3323232 (stretched 19edo diatonic)
* [[Taric]] (8L2s), 2222122221 and the altered MOS pentachordal taric, 2221222221
* Hexawood (6L6s) is a "straddle-3 chromatic scale", constructed by an alternating stack of flat and sharp fifths

== Edostep interpretations ==

==== Edostep interpretations ====
18edo's edostep has the following interpretations in its patent val:

* 25/24 (the difference between 5/4 and 6/5)
* 56/55 (the difference between 11/8 and 7/5)
* 80/77 (the difference between 7/5 and 16/11)
* 36/35 (the difference between 6/5 and 7/6)

== JI approximation ==
18edo is straddle-3 and -7, but inherits 12edo's approximation of 5 and also approximates 11 to a similar degree of accuracy, making 55/32 tuned accurately. Additionally, 7 and 3 are both sharp about 30 cents, meaning 7/6 is tuned well. Oher intervals tuned well include 28/25 and 25/24. Additionally, because 18edo has both the 5 and 7 from 12edo, 7/5 and 10/7 are tempered together ([[jubilic]] temperament) and therefore tuned to 600c (the perfect semioctave). This means that passable tunings of 5:6:7 and its utonal counterpart are available.

Additionally, 18edo's approximate 4:5:6 using the sharp fifth is approximately delta-rational (similar to the case with [[15edo]]), albeit slightly less accurate. It is roughly the isoharmonic JI chord 19:24:29. {{Harmonics in ED|18|11|0}}

== Non-DR theory ==
18edo's best fifth is 733.3 cents, which generates an oneirotonic scale; it can also be used as the double of 9edo, which has an antidiatonic scale.

In oneirotonic, the "thirds" are the same degree as the "major second" and "perfect fourth" respectively, so that any oneirotonic scale always has two distinct "thirds". This follows from the generator being an odd number of steps. In 13edo, the four possible "thirds" collapse to 3 (as the "diminished fourth" and "major third" are identical) but in 18edo (and in 21edo) there are four distinct qualities, here named with [[ADIN|oneirotonic ADIN]]:

* [0 4 11] - "inframinor" (267c)
* [0 5 11] - "nearminor" (333c)
* [0 6 11] - "nearmajor" (400c)
* [0 7 11] - "ultramajor" (467c)

([https://dxinteractive.github.io/xenpaper/#%7B18edo%7D%0A(osc%3Afmtriangle)%0A%5B0_4_11%5D----_._._%5B0_5_11%5D----_._._%5B0_6_11%5D----_._._%5B0_7_11%5D----_._. listen])

The brightest mode of oneirotonic (other than the very brightest, which doesn't have the 733c fifth) has ultramajor and nearmajor. Most modes have ultramajor and inframinor, corresponding to diatonic sus4 and sus2 chords. The darkest mode has nearminor and inframinor.

Alternately, in antidiatonic, 18edo adds to 9edo a "neutral" third, which is the same as the nearminor oneirotonic third. Therefore, there are 5 different qualities of antidiatonic fifth-bounded triad:

* [0 3 10] - "inframinor" (200c)
* [0 4 10] - "farminor" (267c)
* [0 5 10] - "neutral" (333c)
* [0 6 10] - "farmajor" (400c)
* [0 7 10] - "ultramajor" (467c)

([https://dxinteractive.github.io/xenpaper/#%7B18edo%7D%0A(osc%3Afmtriangle)%0A%5B0_3_10%5D--_.._%5B0_4_10%5D--_.._%5B0_5_10%5D--_.._%5B0_6_10%5D--_.._%5B0_7_10%5D--_.._ listen])

These chords also exist in oneirotonic (except for the neutral one), as the antidiatonic fifth is also the oneirotonic major tritone, which appears on 4 different scale degrees.

Additionally, there is the dual-fifth interpretation of 18edo, wherein the basic scale is similar to a 19edo diatonic but with one edostep removed. It is useful to organize the chords based on their appearance on each degree of dual-fifth diatonic.
{| class="wikitable"
|+
!Degree
!LLsLLLm
!LLmLLLs
|-
|1
|[0 6 10] - major antidiatonic
|[0 6 11] - nearmajor
|-
|2
|[0 4 10] - minor antidiatonic
|[0 5 11] - nearminor
|-
|3
|[0 4 10] - minor antidiatonic
|[0 5 11] - nearminor
|-
|4
|[0 6 11] - nearmajor
|[0 6 10] - major antidiatonic
|-
|5
|[0 6 11] - nearmajor
|[0 6 10] - major antidiatonic
|-
|6
|[0 5 11] - nearminor
|[0 4 10] - minor antidiatonic
|-
|7
|[0 5 9] - otonal diminished
|[0 4 9] - utonal diminished
|}
The otonal and utonal diminished triads provide tunings of 5:6:7 and 1/(5:6:7) respectively, despite the fact that 18edo tunes neither 5 nor 7 particularly well (in fact, it has the same mappings as 12edo).

To use a scale which includes neutral harmony, an option would be altering some of the antidiatonic degrees to be neutral. This generates "smitonic" (3-3-2-3-2-3-2) or a MODMOS thereof, which can be thought of as the antidiatonic counterpart of [[mosh]]. The MOS form of smitonic has five of the seven tertian triads neutral; the remaining two are nearmajor and nearminor triads using the oneirotonic fifth. The MODMOS 2-3-3-2-2-3-3 introduces more variation, adding in a major and a minor antidiatonic triad, leaving neutral triads on three of the degrees.

=== Scale Workshop links ===
[https://scaleworkshop.plainsound.org/scale/RBxV9E9cC Smitonic MODMOS] (inverted step pattern of the scale associated with rast maqam)

[https://scaleworkshop.plainsound.org/scale/RBxvWZhiE Smitonic] (neutralized antidiatonic)

[https://scaleworkshop.plainsound.org/scale/RBxzvsm6S Antidiatonic] (inverted step pattern of diatonic)

[https://scaleworkshop.plainsound.org/scale/RBx7WVRn- Oneirotonic]

== DR theory ==
18edo has the following approximate [[DR]] chords (below 10c pairwise logarithmic least-squares error, bounding interval < 1200c, no 1\18 or 17\18):
=== +1+1 ===
* [0 8 14]\18 (error 6.3c)
* [0 6 11]\18 (error 7.3c) - this is the "nearmajor" oneirotonic chord, which behaves like a stretched 4:5:6 in a delta-rational context (a similar chord is [0 5 9] of [[15edo]]).
=== +1+2 ===
* [0 6 15]\18 (error 1.3c)
* [0 3 8]\18 (error 4.8c)
* [0 5 13]\18 (error 8.6c)
=== +2+1 ===
* [0 7 10]\18 (error 6.8c)
* [0 11 15]\18 (error 9.5c)
=== +1+?+1 ===
* [0 4 8 11]\18 (error 0.2c)
* [0 3 11 13]\18 (error 0.2c)
* [0 7 10 15]\18 (error 3.3c)
* [0 6 12 16]\18 (error 5.2c)
* [0 6 11 15]\18 (error 6.8c) (stretched 4:5:6:7)
* [0 5 7 11]\18 (error 7.8c)
* [0 4 7 10]\18 (error 8.6c)
* [0 4 9 12]\18 (error 8.8c)

== Supersets ==
18edo's primes are mostly off by about a twelfth-tone or a sixth-tone. This implies that multiplying it by four to yield [[72edo]] yields an accurate tuning of just intonation. [[36edo]] contains 72edo's 2.3.7 subgroup and is the next level of structural resolution for said subgroup after [[5edo]]. {{Navbox EDO}}
{{Cat|Edos}}

Note entry

2026-03-19T20:00:22Z

Hkm: /* Notation */

{{problematic}}

'''Note entry''' refers to the ways that notes are entered into a digital audio workstation (DAW) when composing. The two most common methods are '''piano roll''' and '''notation'''.

== Piano roll ==
The piano roll can be utilized in several different ways depending on the preferences of the user. It is common for composers to have a maximum number of notes equal to a useful edo, such as 31, 41, 46, or 53. Because there are only 128 MIDI notes, range is a concern for larger edos.
{| class="wikitable"
!Range (Octaves)
!Max Edo
|-
|1
|128
|-
|2
|64
|-
|3
|42
|-
|4
|32
|-
|5
|25
|-
|6
|21
|-
|7
|18
|-
|8
|16
|}
Note that N octaves of range means that each octave-equivalent pitch class occurs exactly N times. The table stops at 8 octaves because that covers the entire usable pitch range, from a bass guitar with extremely heavy strings to a professional glockenspiel.

It is also possible to use midi channels to increase range. This can be done by either assigning them to octave offsets, steps of an edo superset, or a combination of both. The former is available in Surge synth by default, but the others tend to require custom scripts like the in-progress [https://spoogly.website/tools.html TuneLoon].

== Notation ==
MuseScore has a few plugins for microtuning. The one most commonly used is [https://github.com/AzureDevs/XenKit XenKit] (which supports MuseScore 3 and 4), which allows the user to specify an EDO or JI and redefines MuseScore’s built-in accidentals to mean alterations within that tuning system according to standard notation. MuseScore by default does not allow multiple accidentals to be placed on the same note (as would be necessary for JI notation), but XenKit allows these extra accidentals to be applied by writing them as lyrics.
{{Cat|
Core knowledge
Praxis
}}

13edo

2026-03-19T14:48:27Z

Hkm:

'''13edo''', or 13 equal divisions of the octave, is the equal tuning featuring steps of (1200/13) ~= 92.308 cents, 13 of which stack to the octave 2/1. It does not approximate many small prime harmonics well at all, and the JI approximations it does have do not fit very well in a temperament accessed by a particular [[MOS]] scale (they fit better in a [[Glossary#Neji|neji]]), so [[Delta-rational chord|DR]]-based interpretations may be preferred among 13edo users.

13edo's greatest melodic strength is its proximity to 12edo, whose most important effect is providing an [[oneirotonic]] (5L3s, LLsLLsLs) MOS which is a compressed diatonic. A functional system for 13edo oneirotonic is provided below.

== Tuning theory ==
=== Intervals ===
{{Proposed}}
Note: The logic of [[User:Ground|ground]]'s notation is to preserve the diatonic order of nominals for the stacked oneirotonic subfourth generators, with one additional note: BEADGCFX
{| class="wikitable"
|+
!Edostep
!Cents
!Interval region name
!ADIN name (Oneirotonic extension)
!Oneirotonic [https://en.xen.wiki/w/TAMNAMS TAMNAMS] name
!Oneirotonic KISS notation
!Ground's notation (on A = 440 Hz)
!26edo subset notation (on A = 440 Hz)
|-
|0
|0
|Unison
|Unison
|Perfect 0-(oneiro)step (P0oneis)
|1
|A
|A
|-
|1
|92.3
|Minor 2nd
|Minor second
|Minor 1-(oneiro)step (m1oneis)
|1# / 2b
|A# / Cb
|Ax / Bbb
|-
|2
|184.6
|Major 2nd
|Major second
|Major 1-(oneiro)step (M1oneis)
|2
|C
|B
|-
|3
|276.9
|(Sub)minor 3rd
|Minor third
|Minor 2-(oneiro)step (m2oneis)
|3
|B
|Bx / Cb
|-
|4
|369.2
|(Sub)major 3rd
|Major third
|Major 2-(oneiro)step (M2oneis) Diminished 3-(oneiro)step (d3oneis)
|3# / 4b
|B# / Db
|C#
|-
|5
|461.5
|Subfourth
|Fourth
|Perfect 3-(oneiro)step (P3oneis)
|4
|D
|Db
|-
|6
|553.8
|Ultrafourth / Infratritone
|Minor tritone
|Minor 4-(oneiro)step (m4oneis)
|5b
|Fb
|D#
|-
|7
|647.2
|Ultratritone / Infrafifth
|Major tritone
|Major 4-(oneiro)step (M4oneis)
|5
|F
|Eb
|-
|8
|738.5
|Superfifth
|Fifth
|Perfect 5-(oneiro)step (P5oneis)
|6
|E
|E# / Fbb
|-
|9
|830.8
|(Super)minor 6th
|Minor sixth
|Augmented 5-(oneiro)step (A5oneis) Minor 6-(oneiro)step (m6oneis)
|6# / 7b
|E# / Gb
|F
|-
|10
|923.1
|(Super)major 6th
|Major sixth
|Major 6-(oneiro)step (M6oneis)
|7
|G
|Fx / Gbb
|-
|11
|1015.4
|Minor 7th
|Minor seventh
|Minor 7-(oneiro)step (m7oneis)
|8b
|Xb
|G
|-
|12
|1107.7
|Major 7th
|Major seventh
|Major 7-(oneiro)step (M7oneis)
|8
|X
|Gx / Abb
|-
|13
|1200
|Octave
|Octave
|Perfect 8-(oneiro)step (P8oneis)
|1
|A
|A
|}

=== Prime harmonic approximations ===
{{Harmonics in ED|13|23|0}}

=== Edostep interpretations ===
13edo's edostep functions in the 2.9.5.21.11.13.17.19 subgroup as:

* 17/16
* 18/17
* 19/18
* 20/19
* 21/20 (the interval between 10/9 and 7/6)
* 22/21
* 26/25 (the interval between 5/4 and 13/10)
* 55/52 (the interval between 11/8 and 13/10, and between 5/4 and 13/11)
* 128/121 (the interval between 11/8 and 16/11)

=== Harmonic series approximations ===
13edo approximates the following harmonic series chord well (x indicates notes that are harder to approximate):

34:36:38:40:42:x:47:x:52:55:58:61:x:68

This can be derived as follows:
# the quasi-13edo isoharmonic chord 5:9:13:17:21 => 17:18:x:20:21:x:x:x:26:x:x:x:x:34
# the simic sixth chord 17:20:26:29 (+1+2+1) => 17:18:x:20:21:x:x:x:26:x:29:x:x:34
# place 11/8 on harmonic 20 => 34:36:x:40:42:x:x:x:52:55:58:x:x:68
# use halfway harmonics 19 and 47 => 34:36:38:40:42:x:47:x:52:55:58:x:x:68
# 61/52 is .6c off from 3\13 => 34:36:38:40:42:x:47:x:52:55:58:61:x:68

Making an over-17 13edo neji thus requires you to choose those three notes:
* The notes resulting in lowest pairwise error in mode 34 are 44, 49, and 64.
* The closest notes in mode 68 are 89, 99, and 129 (which are significantly more complex).
* A less accurate but lower-complexity neji (limited to oneirotonic) is 22:25:26:29:32:34:38:42:44, so one could specifically choose 44, 50, and 64.

== Jaimbee and Inthar's functional system for 13edo ==
{{Proposed}}

The following system has been developed by Jaimbee and Inthar.

13edo's melodically strongest scale is the oneirotonic MOS (preserving the diatonic property of having at least 2 semitones), so it behooves us to find harmonies that work for it. Since there are certain similarities of oneirotonic to diatonic, we can build off of these similarities to assign functions to oneirotonic degrees.

For a DR-forward framework like this, prefer mellow timbres to bright ones to bring out the DR effect.
=== Note on chord symbols ===
Degrees I, IV (fourth from root), and V (fifth from root) are perfect by default.

Degrees II (second from root), III (third from root), T (tritone from root), VI (sixth from root), and VII (seventh from root) may be written as follows to explicitly indicate the quality of the interval from the root: mII, mIII, mT, mVI, and mVII for minor intervals from the root and MII, MIII, MT, MVI, and MVII for major intervals from the root. When quality is not explicitly indicated, the quality is from the current mode being discussed.

The chord symbols used are ground's system, described in the [[Oneirotonic]] article.

=== Basic chords ===
The most basic chords in this functional harmony system are:
* Tract-major triad 0-4-7\13 (<code>>maj<</code>): A compressed major triad that sounds desaturated and somewhat bittersweet. Somewhat dubiously +1+1. Oneirotonic provides only two of these triads, so alterations are somewhat frequently used to get this triad. The tract-major triad has the following important tetrad supersets:
** 0-2-4-7\13 (<code>>majsus2<</code>): Reinforces the quasi-DR effect with an extra tone; approximately +1+1+2.
** 0-4-7-10\13 (<code>>dom7<</code>): A compressed dominant tetrad; approximately +1+?+1.
** 0-4-7-12\13 (<code>>maj<7</code>): Approximately +1+1+2.
* The simic triad 0-3-8\13 (<code>sim</code>): A bright and brooding if somewhat hollow-sounding minor triad. Approximately 17:20:26 or +1+2. The important supersets are:
** 0-3-8-10 (<code>sim6</code>): Approximately +1+2+1.
** 0-3-8-12 (<code>simmaj7</code>): Approximately +1+2+2.
** 0-3-8-11 (<code>simmin7</code>): Something like a minor 7th tetrad.
** 0-3-8-15 (<code>simadd9</code>)
** 0-3-8-12-15 (<code>simmaj7add9</code>): A concatenation of the minor +1+2 and major +1+1 triads.
* 0-5-9\13 (<code>>IV/I<</code>): A +1+1 triad and a compressed 2nd inversion major triad. Approximately 13:17:21.
** 0-5-7-9: Approximately +2+1+1.
** 0-5-9-12: A tract-major triad on top of a subfourth.
** 0-5-9-12-15
** 0-5-7-9-12-15-17
* 0-5-7\13 (<code>>sus4<</code>): Compressed sus4. Approximately +2+1.
* 0-4-8\13 (<code>>aug<</code>): "Submajor augmented" triad.
* 0-3-6\13 (<code>>dim<</code>): The most diminished-like triad.

=== Functional patterns ===
13edo oneiro enjoys two main (rooted) delta-rational sonorities analogous to major and minor triads: 0-(185)-369-646 ("tract-major triad" or just ">maj<") and 0-277-738-923 ("simic sixth" or "sim6"). One of these chords are on the root in the 6 brightest modes of oneirotonic. In the two darkest modes, I think 0-277-738-1015 or 0-738-1015-277 works well. The chord 0-277-738 will be called "simic", and 0-277-646-923 will be called "tract-minor 7th".

[https://luphoria.com/xenpaper/#(bpm%3A60)(osc%3Asawtooth2)%7B13edo%7D%0A%7Br179hz%7D%0A(env%3A1811)%0A%23_0_2_3_5_7_8_10_12_13%0A%5B%600_%608_3_8_10_'0%5D-%0A%5B%602_5_10_12_'2%5D-%0A%5B%603_5_7_10_'0_'3%5D-%0A%5B%605_8_13_'2_'5%5D-%0A%5B%607_10_13_'3_'7%5D-%0A%5B%608_10_12_'2_'8%5D-%0A%5B%6010_'0_'5_'7_'10%5D-%0A%5B%6012_'2_'5_'8_'12%5D-%0A%5B0_'3_8_10_'13%5D- A progression on the ascending Celephaïsian scale]

[https://luphoria.com/xenpaper/#(bpm%3A60)(osc%3Asawtooth2)%7B13edo%7D%0A%7Br274.988hz%7D%0A(env%3A1811)%0A%23_0_2_3_5_6_8_10_12_13%0A%5B%600_%608_0_3_10_%270%5D-%0A%5B%602_%6010_2_5_10%5D12%0A%5B%603_0_3_6_13_%273%5D-%0A%5B%605_2_8_13_15_%278%5D%275%0A%5B%606_0_3_8_10_%276%5D-%0A%5B%608_2_5_10_%275%5D-%0A%5B%6010_0_6_8_%273%5D-%0A%5B%6012_5_8_12_%272%5D-%0A%5B0_3_8_10_12_13%5D- A progression on the ascending Melodic Mnarian scale]

Adding 923 and 1108 to chords works well, and for jazzy extensions one can add 185, 461, and 646 to the upper octave.

[https://luphoria.com/xenpaper/#(bpm%3A240)(osc%3Asawtooth2)%7B13edo%7D%0A%7Br360hz%7D%0A(env%3A1847)%0A%23_0_2_3_5_6_8_10_11~12_13%0A%5B%600_%608_0%5D_3_6_8-----........%0A%5B%600_%608_0%5D_3_8_10-----.......13%0A%5B%60%608_%608_11%5D_%6011_2_3---6_3-2-3-5-%0A%5B%605_5_8_11_13_8%5D-_11_13----_13_'2------%0A%5B%600_%608_0%5D_3_6_8-----........%0A%5B%600_%608_0%5D_3_8_10-----.....'3'2'0%0A%5B%603_3_6_8_0_11%5D-13_%5B%602_5_8%5D-_11-_8_%5B0_10%5D-_%5B%6011_8%5D_%5B%6010_6%5D-%5B%6011_8%5D_%5B%6010_6%5D_%5B%608_5%5D%0A%5B%603_%606_%6011_3%5D-_%5B%602_5%5D-_%5B%600_3%5D-.._%0A%5B%608_%608_%6010_%6012_2%5D_%608_%6010_%6012---- A Mnarian loop with an &8 leading tone at the end]

[https://luphoria.com/xenpaper/#(bpm%3A30)(osc%3Asawtooth2)%0A%7Br200Hz%7D%7B13edo%7D%0A(env%3A1846)%0A%5B0_4_7_10_%272_%276_%279_%2712%5D-.._%23_Dylydian%0A%5B0_5_9_%270_%272_%274_%277_%2710_%2712%5D-.._%23_Dylathian%0A%5B0_5_8_10_%270_%272_%274_%277_%2712%5D-.._%23_Illarnekian%0A%5B0_3_8_10_%272_%275_%277_%2712%5D-.._%23_Celepha%C3%AFsian%0A%5B0_5_9_%273_%277_%2710_%2712_%27%272%5D-.._%23_Celdorian%0A%5B0_3_6_8_%2711_%272_%275_10%5D-.._%23_Mnarian%0A%5B0_3_6_8_10_%272_%275_%2712%5D-.._%23_Mnionian Some motherchords of oneiro modes]
=== Functional chords on each degree ===
Celephaisian
* I: sim6
* II: sim6
* III: >maj<
* IV: sim6, >min7<
* T: sim, minor 4ms, minor 6ms, minor 7ms
* V: >maj<
* VI: sim6
* VII: sim, minor 4ms, minor 6ms, minor 7ms

=== Progressions ===
Common motions:
* I(>maj< or sim6) → MIIsim6
* I(>maj< or sim6) → IVsim6 (when ending on 0d this sounds like diatonic V to I)
* I(>maj< or sim6) → mVIsim6
* I>maj< → V>maj<, Isim6 → V(>maj< or sim6) (when ending on 0d this is a "dominant to tonic" motion)
* I>maj< → MIII>maj<
* MIIsim6 → mII>maj< → I>maj<

=== Functional harmony ===
Modes can be grouped by their functional properties.
* Dual-fifth: Illarnekian, Celephaïsian, Ultharian
* Dual-fourth: Mnarian, Kadathian, Hlanithian
* Tract-major chord on root: Dylathian, Illarnekian
* Simic sixth chord on root: Celephaïsian, Ultharian, Mnarian, Kadathian
* Lower leading tone: Dylathian, Illarnekian, Celephaïsian
* "Neoclassical functional modes" (loose grouping): Dylathian, Illarnekian, Celephaïsian, Ultharian
* Upper leading tone: Kadathian, Hlanithian, Sarnathian,
* Minor 6-mosstep: Hlanithian, Sarnathian,
* 0 462 831 delta-rational chord on root: Dylathian, Dylydian, Hlanithian,
* "Dorian-like", i.e. no leading tone, 5d is minor, and 6d is major: Ultharian, Mnarian
* 7d is minor: Kadathian, Hlanithian
We'll call degrees that don't have a >maj< or sim6 chord ''dissonant degrees'' (keeping in mind that dissonance is a feature a chord has in a musical language rather than a purely psychoacoustic property).
==== Dylathian ====
The below uses I, II, III, IV, T, V, VI, VII degrees/functions and ground's notation for oneirotonic nominals. Interval names are in ADIN.

In Dylathian, we find tract-major chords on the I and IV degrees, while simic chords appear on
the II, T, VI, and VII degrees. For the III and V degrees, you get a chord of edo steps 0-3-9-11, which is the third type of DR tetrad, which could be viewed as an inversion of the tract-dominant tetrad. Alternatively, you could also play a DR chord of scale degrees 0-5-9 on the third degree, and in some contexts it may be favorable (see below).

For each of these chords, we can associate functions with them. The simplest of these relationships is between
the root tract-major chord and the tract-major chord on the perfect fourth. By adding octaves on certain notes, we can recreate the familiar dominant cadence from diatonic, only now on the IV rather than the V. The most simple of these progressions would look something like this (in ground's notation):
# B-D-F-G-F
# E-G-X-C-D
# B-D-F-G-F
Or, in 13edo steps:
# 0-2-4-7-17
# 5-7-9-12-18
# 0-2-4-7-17
In this cadence, the fifth on E is so narrow that it creates a leading tone relative to the root, and by playing the
octaves above E, you can create a minor tritone that wants to resolve inwards to the tract-major chord on the root.
The presence of the octaves above the major third helps drive this resolution, but can be omitted.
Another neat effect is that given the dominant is now on the IV, then the II simic sixth chord would be exactly
halfway to the dominant, making it the mediant. It also has a much nicer simic sixth chord on it compared to the III
1st inversion tract-major chord, making it more akin to how the mediant works in diatonic.

We can also relate other chords to the dominant, mediant and tonic. The relative minor is more or less exactly
analogous to diatonic, being a minor third below the tonic (in the case of B Dylathian, it would be A
Celephaïsian, the minor VI). The mediant can also function as a secondary dominant for resolutions to the
relative minor; the highest note in the II minor chord is one semitone below the minor third of the VI minor
chord. By playing the octave above certain notes, resolving between the two modes is pretty simple.
# B-D-F-G-B
# `D-`E-`A-`C-C
# `A-`B-`E-`G-B
In 13edo steps:
# 0-2-4-7-13
# `2-`5-`10-`12-12
# `10-0-5-7-13
The ` denotes playing an octave lower than the root.

The II (D minor) and VI (A minor) would probably sound the smoothest when played in a lower register than the
tonic (B major) as notated, but if you want to move upwards from the root it still works.
Resolving from the relative minor (A minor) to the tonic (B major) is a pretty weak but still usable resolution. Another neat resolution is moving from the III (inverted major) to the dominant IV.

If you play the 0-5-9 chord on the third degree, the lowest note will be a semitone lower than the
lowest in the dominant, and the highest note will be a semitone higher than the highest in the dominant. By
either extending the 0-5-9 chord to 0-3-5-9, or simplifying the dominant chord to a 0-4-7 chord, you can drive this resolution very powerfully, and this could either create a chain of strong resolutions going iii (inverted major)-IV-I, or it could help drive resolutions to the Ilarnekian mode above (in this case, E Ilarnekian).

Technically you wouldn't have to extend or simplify any of these chords, but the triad next to all the tetrads feels
somewhat empty. All in all, using this technique you could probably simplify all the tetrads down to 0-4-7 and 0-3-8
for major and minor, respectively. These would help since the 0-3-5-9 chord doesn't have much of a DR effect, while the simplified major and minor still have a DR effect, though a bit weaker than the tetrads. The vi inverted major (9-12-18-20) chord also has some neat features, as it functions as an inversion of the dominant IV chord. It also doesn't need any extensions with octaves to work well unlike the dominant chord, so it could be seen as a more tense version of dominant. Since it also drives the resolution up by a minor third, the same tetrad on the III could be used to drive a resolution to a major V, helping to shift the key center from B to G#. If done twice, this resolution can shift your key center up a minor third from B -> G# -> F#, which gives the progression a really jazzy feel.

The only chord we haven't covered now would be the minor T (7-10-15-17). This chord has a much
weaker relationship to the other chords, so it doesn't have any strong directionality. However, it does share
some notes with a few important chords, notably the I chord and the relative minor on the VI. A resolution to
either of these will be similarly strong, that is to say, not very strong. In this case it could also be seen as a
secondary mediant which is not the relative minor, about halfway between the I chord and the VII chord a
octave above it, and either of these resolutions would probably sound fine in most contexts. This gives it a role
completely unlike any of the functions in traditional diatonic. It also works pretty well as a setup for the V inverted major, so in a progression it can help add some flair or beef to the resolution.

==== Ilarnekian ====
To start with the basics, Ilarnekian is just Dylathian with a flattened 6th.
In E Ilarnekian, you'd get:
E G X A C B D F E

With Ilarnekian being the second major mode (after Dylathian), we'd get the same I chord, E>maj< One of the most immediate effects we'd see, however, is that the dominant IV is now IV>min7<. It still can function as a dominant, though only with the added octave above the 4th, and slightly weaker than the Dylathian dominant cadence. Another interesting thing is that the IV-I cadence is now simultaneously a minor plagal and a dominant cadence, radically different from anything in diatonic.

The II chord is again simic sixth, with it now driving a secondary resolution to D Ultharian instead of D Celephaïsian. The III inverted major chord is also still first inversion tract-major, and also still drives a pretty good resolution up to the simic sixth IV, though admittedly weaker than Dylathian.

The T chord is still simic sixth, and still functions as a secondary mediant.

It gets interesting again when looking at the tract-major V chord.

By playing the V lower than the tonic and playing the octave above the root of the V chord, you get an entirely new approach to a dominant chord. The third of the V is one semitone below the tonic, and the octave above the root is one semitone above the fifth of the tonic chord. This drives a very strong inward resolution that resembles dominant in diatonic slightly more than the dominant IV chord in Dylathian, and a lot more than the tract-minor IV chord in Ilarnekian.
It would look something like this:
* E G X C
* `B `D `F G B
* E G X C

In 13edo steps:
* 0-2-4-7
* `8-`10-`12-2-8
* 0-2-4-7
with ` again notating playing an octave lower than the starting chord.

Moving on, the VI chord would be simic sixth driving the resolution to the IV simic sixth,
in the same way Dylathian's II simic sixth drives the resolution to the relative minor. It would also function as the
Ilarnekian relative minor, in this case D Ultharian. The vii chord would be an inverted major chord, and would drive a resolution to the tonic pretty well. This comes from the fact that the 0-3-9-11 chord would have the minor third become the major second of the tonic, the root move up a semitone to the tonic, and the perfect fifth move down a semitone to become the tritone of the tonic
chord. The minor sixth in the chord could be omitted to make the resolution stronger, but the chord would sound much more dissonant.

==== Celephaïsian ====
Functional chords on each degree:
* I: sim6
* II: sim6
* III: >maj<
* IV: sim6
* T: sim; >maj<₁ (first inversion >maj<)
* VI: >maj<
* VII: sim6
* VIII: sim; >maj<₁
The main resolving degrees (analogues to dominant in diatonic) are IV and V because of their leading tones.

Progressions:
* Isim6 IIsim6 IV(sim6 or >min7<) Isim6
* Isim6 IIsim6 VIsim6 IV(sim6 or >min7<) Isim6
* Isim6 VIsim6 IV(sim6 or >min7<) Isim6
* Isim6 III>maj< IV(sim6 or >min7<) Isim6
* Isim6 Tsim7 V>maj< IV(sim6 or >min7<) Isim6
* Isim6 VIsim6 V>maj< Isim6

Secondary modes:
* IV Ultharian
* III Dylathian
* V Illarnekian

== Multiples ==

=== 26edo ===
:''Main article: [[26edo]]''

=== 39edo ===
39edo is a Supra (2.3.11[17 & 22]) diatonic tuning which has good 11/8 and 9/7 approximations in the mosdiatonic scale. One may favor 39edo over harder Archy tunings for the larger diatonic semitone size.
{{Harmonics in ED|39|31|0}}

=== 65edo ===
65edo is notable as the intersection of [[Schismic]] and [[Wurschmidt]]. It is a strong 2.3.5.11.19.23.31.47.49 system.
{{Harmonics in ED|65|47|0}}

===104edo===
''See [[26edo#104edo]].''

===130edo===
''See [[26edo#130edo]].''

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

13edo

2026-03-19T14:48:12Z

Hkm:

'''13edo''', or 13 equal divisions of the octave, is the equal tuning featuring steps of (1200/13) ~= 92.308 cents, 13 of which stack to the octave 2/1. It does not approximate many small prime harmonics well at all, and the JI approximations it does have do not fit very well in a temperament accessed by a particular scale (they fit better in a [[Glossary#Neji|neji]]), so [[Delta-rational chord|DR]]-based interpretations may be preferred among 13edo users.

13edo's greatest melodic strength is its proximity to 12edo, whose most important effect is providing an [[oneirotonic]] (5L3s, LLsLLsLs) MOS which is a compressed diatonic. A functional system for 13edo oneirotonic is provided below.

== Tuning theory ==
=== Intervals ===
{{Proposed}}
Note: The logic of [[User:Ground|ground]]'s notation is to preserve the diatonic order of nominals for the stacked oneirotonic subfourth generators, with one additional note: BEADGCFX
{| class="wikitable"
|+
!Edostep
!Cents
!Interval region name
!ADIN name (Oneirotonic extension)
!Oneirotonic [https://en.xen.wiki/w/TAMNAMS TAMNAMS] name
!Oneirotonic KISS notation
!Ground's notation (on A = 440 Hz)
!26edo subset notation (on A = 440 Hz)
|-
|0
|0
|Unison
|Unison
|Perfect 0-(oneiro)step (P0oneis)
|1
|A
|A
|-
|1
|92.3
|Minor 2nd
|Minor second
|Minor 1-(oneiro)step (m1oneis)
|1# / 2b
|A# / Cb
|Ax / Bbb
|-
|2
|184.6
|Major 2nd
|Major second
|Major 1-(oneiro)step (M1oneis)
|2
|C
|B
|-
|3
|276.9
|(Sub)minor 3rd
|Minor third
|Minor 2-(oneiro)step (m2oneis)
|3
|B
|Bx / Cb
|-
|4
|369.2
|(Sub)major 3rd
|Major third
|Major 2-(oneiro)step (M2oneis) Diminished 3-(oneiro)step (d3oneis)
|3# / 4b
|B# / Db
|C#
|-
|5
|461.5
|Subfourth
|Fourth
|Perfect 3-(oneiro)step (P3oneis)
|4
|D
|Db
|-
|6
|553.8
|Ultrafourth / Infratritone
|Minor tritone
|Minor 4-(oneiro)step (m4oneis)
|5b
|Fb
|D#
|-
|7
|647.2
|Ultratritone / Infrafifth
|Major tritone
|Major 4-(oneiro)step (M4oneis)
|5
|F
|Eb
|-
|8
|738.5
|Superfifth
|Fifth
|Perfect 5-(oneiro)step (P5oneis)
|6
|E
|E# / Fbb
|-
|9
|830.8
|(Super)minor 6th
|Minor sixth
|Augmented 5-(oneiro)step (A5oneis) Minor 6-(oneiro)step (m6oneis)
|6# / 7b
|E# / Gb
|F
|-
|10
|923.1
|(Super)major 6th
|Major sixth
|Major 6-(oneiro)step (M6oneis)
|7
|G
|Fx / Gbb
|-
|11
|1015.4
|Minor 7th
|Minor seventh
|Minor 7-(oneiro)step (m7oneis)
|8b
|Xb
|G
|-
|12
|1107.7
|Major 7th
|Major seventh
|Major 7-(oneiro)step (M7oneis)
|8
|X
|Gx / Abb
|-
|13
|1200
|Octave
|Octave
|Perfect 8-(oneiro)step (P8oneis)
|1
|A
|A
|}

=== Prime harmonic approximations ===
{{Harmonics in ED|13|23|0}}

=== Edostep interpretations ===
13edo's edostep functions in the 2.9.5.21.11.13.17.19 subgroup as:

* 17/16
* 18/17
* 19/18
* 20/19
* 21/20 (the interval between 10/9 and 7/6)
* 22/21
* 26/25 (the interval between 5/4 and 13/10)
* 55/52 (the interval between 11/8 and 13/10, and between 5/4 and 13/11)
* 128/121 (the interval between 11/8 and 16/11)

=== Harmonic series approximations ===
13edo approximates the following harmonic series chord well (x indicates notes that are harder to approximate):

34:36:38:40:42:x:47:x:52:55:58:61:x:68

This can be derived as follows:
# the quasi-13edo isoharmonic chord 5:9:13:17:21 => 17:18:x:20:21:x:x:x:26:x:x:x:x:34
# the simic sixth chord 17:20:26:29 (+1+2+1) => 17:18:x:20:21:x:x:x:26:x:29:x:x:34
# place 11/8 on harmonic 20 => 34:36:x:40:42:x:x:x:52:55:58:x:x:68
# use halfway harmonics 19 and 47 => 34:36:38:40:42:x:47:x:52:55:58:x:x:68
# 61/52 is .6c off from 3\13 => 34:36:38:40:42:x:47:x:52:55:58:61:x:68

Making an over-17 13edo neji thus requires you to choose those three notes:
* The notes resulting in lowest pairwise error in mode 34 are 44, 49, and 64.
* The closest notes in mode 68 are 89, 99, and 129 (which are significantly more complex).
* A less accurate but lower-complexity neji (limited to oneirotonic) is 22:25:26:29:32:34:38:42:44, so one could specifically choose 44, 50, and 64.

== Jaimbee and Inthar's functional system for 13edo ==
{{Proposed}}

The following system has been developed by Jaimbee and Inthar.

13edo's melodically strongest scale is the oneirotonic MOS (preserving the diatonic property of having at least 2 semitones), so it behooves us to find harmonies that work for it. Since there are certain similarities of oneirotonic to diatonic, we can build off of these similarities to assign functions to oneirotonic degrees.

For a DR-forward framework like this, prefer mellow timbres to bright ones to bring out the DR effect.
=== Note on chord symbols ===
Degrees I, IV (fourth from root), and V (fifth from root) are perfect by default.

Degrees II (second from root), III (third from root), T (tritone from root), VI (sixth from root), and VII (seventh from root) may be written as follows to explicitly indicate the quality of the interval from the root: mII, mIII, mT, mVI, and mVII for minor intervals from the root and MII, MIII, MT, MVI, and MVII for major intervals from the root. When quality is not explicitly indicated, the quality is from the current mode being discussed.

The chord symbols used are ground's system, described in the [[Oneirotonic]] article.

=== Basic chords ===
The most basic chords in this functional harmony system are:
* Tract-major triad 0-4-7\13 (<code>>maj<</code>): A compressed major triad that sounds desaturated and somewhat bittersweet. Somewhat dubiously +1+1. Oneirotonic provides only two of these triads, so alterations are somewhat frequently used to get this triad. The tract-major triad has the following important tetrad supersets:
** 0-2-4-7\13 (<code>>majsus2<</code>): Reinforces the quasi-DR effect with an extra tone; approximately +1+1+2.
** 0-4-7-10\13 (<code>>dom7<</code>): A compressed dominant tetrad; approximately +1+?+1.
** 0-4-7-12\13 (<code>>maj<7</code>): Approximately +1+1+2.
* The simic triad 0-3-8\13 (<code>sim</code>): A bright and brooding if somewhat hollow-sounding minor triad. Approximately 17:20:26 or +1+2. The important supersets are:
** 0-3-8-10 (<code>sim6</code>): Approximately +1+2+1.
** 0-3-8-12 (<code>simmaj7</code>): Approximately +1+2+2.
** 0-3-8-11 (<code>simmin7</code>): Something like a minor 7th tetrad.
** 0-3-8-15 (<code>simadd9</code>)
** 0-3-8-12-15 (<code>simmaj7add9</code>): A concatenation of the minor +1+2 and major +1+1 triads.
* 0-5-9\13 (<code>>IV/I<</code>): A +1+1 triad and a compressed 2nd inversion major triad. Approximately 13:17:21.
** 0-5-7-9: Approximately +2+1+1.
** 0-5-9-12: A tract-major triad on top of a subfourth.
** 0-5-9-12-15
** 0-5-7-9-12-15-17
* 0-5-7\13 (<code>>sus4<</code>): Compressed sus4. Approximately +2+1.
* 0-4-8\13 (<code>>aug<</code>): "Submajor augmented" triad.
* 0-3-6\13 (<code>>dim<</code>): The most diminished-like triad.

=== Functional patterns ===
13edo oneiro enjoys two main (rooted) delta-rational sonorities analogous to major and minor triads: 0-(185)-369-646 ("tract-major triad" or just ">maj<") and 0-277-738-923 ("simic sixth" or "sim6"). One of these chords are on the root in the 6 brightest modes of oneirotonic. In the two darkest modes, I think 0-277-738-1015 or 0-738-1015-277 works well. The chord 0-277-738 will be called "simic", and 0-277-646-923 will be called "tract-minor 7th".

[https://luphoria.com/xenpaper/#(bpm%3A60)(osc%3Asawtooth2)%7B13edo%7D%0A%7Br179hz%7D%0A(env%3A1811)%0A%23_0_2_3_5_7_8_10_12_13%0A%5B%600_%608_3_8_10_'0%5D-%0A%5B%602_5_10_12_'2%5D-%0A%5B%603_5_7_10_'0_'3%5D-%0A%5B%605_8_13_'2_'5%5D-%0A%5B%607_10_13_'3_'7%5D-%0A%5B%608_10_12_'2_'8%5D-%0A%5B%6010_'0_'5_'7_'10%5D-%0A%5B%6012_'2_'5_'8_'12%5D-%0A%5B0_'3_8_10_'13%5D- A progression on the ascending Celephaïsian scale]

[https://luphoria.com/xenpaper/#(bpm%3A60)(osc%3Asawtooth2)%7B13edo%7D%0A%7Br274.988hz%7D%0A(env%3A1811)%0A%23_0_2_3_5_6_8_10_12_13%0A%5B%600_%608_0_3_10_%270%5D-%0A%5B%602_%6010_2_5_10%5D12%0A%5B%603_0_3_6_13_%273%5D-%0A%5B%605_2_8_13_15_%278%5D%275%0A%5B%606_0_3_8_10_%276%5D-%0A%5B%608_2_5_10_%275%5D-%0A%5B%6010_0_6_8_%273%5D-%0A%5B%6012_5_8_12_%272%5D-%0A%5B0_3_8_10_12_13%5D- A progression on the ascending Melodic Mnarian scale]

Adding 923 and 1108 to chords works well, and for jazzy extensions one can add 185, 461, and 646 to the upper octave.

[https://luphoria.com/xenpaper/#(bpm%3A240)(osc%3Asawtooth2)%7B13edo%7D%0A%7Br360hz%7D%0A(env%3A1847)%0A%23_0_2_3_5_6_8_10_11~12_13%0A%5B%600_%608_0%5D_3_6_8-----........%0A%5B%600_%608_0%5D_3_8_10-----.......13%0A%5B%60%608_%608_11%5D_%6011_2_3---6_3-2-3-5-%0A%5B%605_5_8_11_13_8%5D-_11_13----_13_'2------%0A%5B%600_%608_0%5D_3_6_8-----........%0A%5B%600_%608_0%5D_3_8_10-----.....'3'2'0%0A%5B%603_3_6_8_0_11%5D-13_%5B%602_5_8%5D-_11-_8_%5B0_10%5D-_%5B%6011_8%5D_%5B%6010_6%5D-%5B%6011_8%5D_%5B%6010_6%5D_%5B%608_5%5D%0A%5B%603_%606_%6011_3%5D-_%5B%602_5%5D-_%5B%600_3%5D-.._%0A%5B%608_%608_%6010_%6012_2%5D_%608_%6010_%6012---- A Mnarian loop with an &8 leading tone at the end]

[https://luphoria.com/xenpaper/#(bpm%3A30)(osc%3Asawtooth2)%0A%7Br200Hz%7D%7B13edo%7D%0A(env%3A1846)%0A%5B0_4_7_10_%272_%276_%279_%2712%5D-.._%23_Dylydian%0A%5B0_5_9_%270_%272_%274_%277_%2710_%2712%5D-.._%23_Dylathian%0A%5B0_5_8_10_%270_%272_%274_%277_%2712%5D-.._%23_Illarnekian%0A%5B0_3_8_10_%272_%275_%277_%2712%5D-.._%23_Celepha%C3%AFsian%0A%5B0_5_9_%273_%277_%2710_%2712_%27%272%5D-.._%23_Celdorian%0A%5B0_3_6_8_%2711_%272_%275_10%5D-.._%23_Mnarian%0A%5B0_3_6_8_10_%272_%275_%2712%5D-.._%23_Mnionian Some motherchords of oneiro modes]
=== Functional chords on each degree ===
Celephaisian
* I: sim6
* II: sim6
* III: >maj<
* IV: sim6, >min7<
* T: sim, minor 4ms, minor 6ms, minor 7ms
* V: >maj<
* VI: sim6
* VII: sim, minor 4ms, minor 6ms, minor 7ms

=== Progressions ===
Common motions:
* I(>maj< or sim6) → MIIsim6
* I(>maj< or sim6) → IVsim6 (when ending on 0d this sounds like diatonic V to I)
* I(>maj< or sim6) → mVIsim6
* I>maj< → V>maj<, Isim6 → V(>maj< or sim6) (when ending on 0d this is a "dominant to tonic" motion)
* I>maj< → MIII>maj<
* MIIsim6 → mII>maj< → I>maj<

=== Functional harmony ===
Modes can be grouped by their functional properties.
* Dual-fifth: Illarnekian, Celephaïsian, Ultharian
* Dual-fourth: Mnarian, Kadathian, Hlanithian
* Tract-major chord on root: Dylathian, Illarnekian
* Simic sixth chord on root: Celephaïsian, Ultharian, Mnarian, Kadathian
* Lower leading tone: Dylathian, Illarnekian, Celephaïsian
* "Neoclassical functional modes" (loose grouping): Dylathian, Illarnekian, Celephaïsian, Ultharian
* Upper leading tone: Kadathian, Hlanithian, Sarnathian,
* Minor 6-mosstep: Hlanithian, Sarnathian,
* 0 462 831 delta-rational chord on root: Dylathian, Dylydian, Hlanithian,
* "Dorian-like", i.e. no leading tone, 5d is minor, and 6d is major: Ultharian, Mnarian
* 7d is minor: Kadathian, Hlanithian
We'll call degrees that don't have a >maj< or sim6 chord ''dissonant degrees'' (keeping in mind that dissonance is a feature a chord has in a musical language rather than a purely psychoacoustic property).
==== Dylathian ====
The below uses I, II, III, IV, T, V, VI, VII degrees/functions and ground's notation for oneirotonic nominals. Interval names are in ADIN.

In Dylathian, we find tract-major chords on the I and IV degrees, while simic chords appear on
the II, T, VI, and VII degrees. For the III and V degrees, you get a chord of edo steps 0-3-9-11, which is the third type of DR tetrad, which could be viewed as an inversion of the tract-dominant tetrad. Alternatively, you could also play a DR chord of scale degrees 0-5-9 on the third degree, and in some contexts it may be favorable (see below).

For each of these chords, we can associate functions with them. The simplest of these relationships is between
the root tract-major chord and the tract-major chord on the perfect fourth. By adding octaves on certain notes, we can recreate the familiar dominant cadence from diatonic, only now on the IV rather than the V. The most simple of these progressions would look something like this (in ground's notation):
# B-D-F-G-F
# E-G-X-C-D
# B-D-F-G-F
Or, in 13edo steps:
# 0-2-4-7-17
# 5-7-9-12-18
# 0-2-4-7-17
In this cadence, the fifth on E is so narrow that it creates a leading tone relative to the root, and by playing the
octaves above E, you can create a minor tritone that wants to resolve inwards to the tract-major chord on the root.
The presence of the octaves above the major third helps drive this resolution, but can be omitted.
Another neat effect is that given the dominant is now on the IV, then the II simic sixth chord would be exactly
halfway to the dominant, making it the mediant. It also has a much nicer simic sixth chord on it compared to the III
1st inversion tract-major chord, making it more akin to how the mediant works in diatonic.

We can also relate other chords to the dominant, mediant and tonic. The relative minor is more or less exactly
analogous to diatonic, being a minor third below the tonic (in the case of B Dylathian, it would be A
Celephaïsian, the minor VI). The mediant can also function as a secondary dominant for resolutions to the
relative minor; the highest note in the II minor chord is one semitone below the minor third of the VI minor
chord. By playing the octave above certain notes, resolving between the two modes is pretty simple.
# B-D-F-G-B
# `D-`E-`A-`C-C
# `A-`B-`E-`G-B
In 13edo steps:
# 0-2-4-7-13
# `2-`5-`10-`12-12
# `10-0-5-7-13
The ` denotes playing an octave lower than the root.

The II (D minor) and VI (A minor) would probably sound the smoothest when played in a lower register than the
tonic (B major) as notated, but if you want to move upwards from the root it still works.
Resolving from the relative minor (A minor) to the tonic (B major) is a pretty weak but still usable resolution. Another neat resolution is moving from the III (inverted major) to the dominant IV.

If you play the 0-5-9 chord on the third degree, the lowest note will be a semitone lower than the
lowest in the dominant, and the highest note will be a semitone higher than the highest in the dominant. By
either extending the 0-5-9 chord to 0-3-5-9, or simplifying the dominant chord to a 0-4-7 chord, you can drive this resolution very powerfully, and this could either create a chain of strong resolutions going iii (inverted major)-IV-I, or it could help drive resolutions to the Ilarnekian mode above (in this case, E Ilarnekian).

Technically you wouldn't have to extend or simplify any of these chords, but the triad next to all the tetrads feels
somewhat empty. All in all, using this technique you could probably simplify all the tetrads down to 0-4-7 and 0-3-8
for major and minor, respectively. These would help since the 0-3-5-9 chord doesn't have much of a DR effect, while the simplified major and minor still have a DR effect, though a bit weaker than the tetrads. The vi inverted major (9-12-18-20) chord also has some neat features, as it functions as an inversion of the dominant IV chord. It also doesn't need any extensions with octaves to work well unlike the dominant chord, so it could be seen as a more tense version of dominant. Since it also drives the resolution up by a minor third, the same tetrad on the III could be used to drive a resolution to a major V, helping to shift the key center from B to G#. If done twice, this resolution can shift your key center up a minor third from B -> G# -> F#, which gives the progression a really jazzy feel.

The only chord we haven't covered now would be the minor T (7-10-15-17). This chord has a much
weaker relationship to the other chords, so it doesn't have any strong directionality. However, it does share
some notes with a few important chords, notably the I chord and the relative minor on the VI. A resolution to
either of these will be similarly strong, that is to say, not very strong. In this case it could also be seen as a
secondary mediant which is not the relative minor, about halfway between the I chord and the VII chord a
octave above it, and either of these resolutions would probably sound fine in most contexts. This gives it a role
completely unlike any of the functions in traditional diatonic. It also works pretty well as a setup for the V inverted major, so in a progression it can help add some flair or beef to the resolution.

==== Ilarnekian ====
To start with the basics, Ilarnekian is just Dylathian with a flattened 6th.
In E Ilarnekian, you'd get:
E G X A C B D F E

With Ilarnekian being the second major mode (after Dylathian), we'd get the same I chord, E>maj< One of the most immediate effects we'd see, however, is that the dominant IV is now IV>min7<. It still can function as a dominant, though only with the added octave above the 4th, and slightly weaker than the Dylathian dominant cadence. Another interesting thing is that the IV-I cadence is now simultaneously a minor plagal and a dominant cadence, radically different from anything in diatonic.

The II chord is again simic sixth, with it now driving a secondary resolution to D Ultharian instead of D Celephaïsian. The III inverted major chord is also still first inversion tract-major, and also still drives a pretty good resolution up to the simic sixth IV, though admittedly weaker than Dylathian.

The T chord is still simic sixth, and still functions as a secondary mediant.

It gets interesting again when looking at the tract-major V chord.

By playing the V lower than the tonic and playing the octave above the root of the V chord, you get an entirely new approach to a dominant chord. The third of the V is one semitone below the tonic, and the octave above the root is one semitone above the fifth of the tonic chord. This drives a very strong inward resolution that resembles dominant in diatonic slightly more than the dominant IV chord in Dylathian, and a lot more than the tract-minor IV chord in Ilarnekian.
It would look something like this:
* E G X C
* `B `D `F G B
* E G X C

In 13edo steps:
* 0-2-4-7
* `8-`10-`12-2-8
* 0-2-4-7
with ` again notating playing an octave lower than the starting chord.

Moving on, the VI chord would be simic sixth driving the resolution to the IV simic sixth,
in the same way Dylathian's II simic sixth drives the resolution to the relative minor. It would also function as the
Ilarnekian relative minor, in this case D Ultharian. The vii chord would be an inverted major chord, and would drive a resolution to the tonic pretty well. This comes from the fact that the 0-3-9-11 chord would have the minor third become the major second of the tonic, the root move up a semitone to the tonic, and the perfect fifth move down a semitone to become the tritone of the tonic
chord. The minor sixth in the chord could be omitted to make the resolution stronger, but the chord would sound much more dissonant.

==== Celephaïsian ====
Functional chords on each degree:
* I: sim6
* II: sim6
* III: >maj<
* IV: sim6
* T: sim; >maj<₁ (first inversion >maj<)
* VI: >maj<
* VII: sim6
* VIII: sim; >maj<₁
The main resolving degrees (analogues to dominant in diatonic) are IV and V because of their leading tones.

Progressions:
* Isim6 IIsim6 IV(sim6 or >min7<) Isim6
* Isim6 IIsim6 VIsim6 IV(sim6 or >min7<) Isim6
* Isim6 VIsim6 IV(sim6 or >min7<) Isim6
* Isim6 III>maj< IV(sim6 or >min7<) Isim6
* Isim6 Tsim7 V>maj< IV(sim6 or >min7<) Isim6
* Isim6 VIsim6 V>maj< Isim6

Secondary modes:
* IV Ultharian
* III Dylathian
* V Illarnekian

== Multiples ==

=== 26edo ===
:''Main article: [[26edo]]''

=== 39edo ===
39edo is a Supra (2.3.11[17 & 22]) diatonic tuning which has good 11/8 and 9/7 approximations in the mosdiatonic scale. One may favor 39edo over harder Archy tunings for the larger diatonic semitone size.
{{Harmonics in ED|39|31|0}}

=== 65edo ===
65edo is notable as the intersection of [[Schismic]] and [[Wurschmidt]]. It is a strong 2.3.5.11.19.23.31.47.49 system.
{{Harmonics in ED|65|47|0}}

===104edo===
''See [[26edo#104edo]].''

===130edo===
''See [[26edo#130edo]].''

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

22edo

2026-02-13T20:46:20Z

Hkm: probably good for seo

'''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.

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 include 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.

== General theory ==
=== 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.

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.

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}}

=== 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]]).

=== Intervals and notation ===
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).

The "native-fifths" system mentioned here is the most commonly used system, and the one that most microtonal notation systems support by default. It is derived through stacking 22edo's tempered version of 3/2 and assigning names accordingly. As a result, the nominals (C, D, E, F, G, ...) follow the diatonic MOS, where the distances between the large steps (C-D, D-E, F-G, G-A, A-B) are 4 EDO steps, and the distances between the small steps (E-F, B-C) are 1 EDO step. Therefore, a sharp corresponds to +3 EDO steps while a flat corresponds to -3 (representing the diatonic chroma in each case). Each sharp or flat can be split into three distinct notes, so we use the accidental ^ to raise by one EDO step and v to lower by one EDO step.

In the other proposed notation systems aside from native fifths, a sharp corresponds to +1 EDO step, while a flat corresponds to -1. The Zarlino notation here uses the [[ternary]] Zarlino scale (see [[#Zarlino diatonic]]), or Ptolemy's intense diatonic, as its basic scale (which prioritizes the 5-limit, whereas native fifths prioritize 2.3.7), and uses to its advantage the fact that one step of 22edo maps to both 25/24 and 81/80 (a property of Porcupine temperament). Pajara uses the 10-note Pajara scale (see [[#Pajara]]) as its basis, which is generated by a perfect fifth but splits the octave in two to reach intervals of the full 7-limit relatively easily; the scale has four 2-step and one 3-step intervals per half-octave, which differ in size by a diatonic minor second, which is 1 step in 22edo.

The ADIN system uses the labels "nearminor" and "nearmajor" for intervals that 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. Note also that as "subminor" and "supermajor" intervals are the minor and major intervals of the diatonic MOS in 22edo, unqualified "major" and "minor" should by default refer to these.

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
|55
|25/24, 28/27, ['''33/32'''], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|164
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218
|8/7, '''9/8''', 17/15
|D
|D#
|2
|(Super)major second
|-
|5
|273
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327
|6/5, 11/9, 17/14
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|382
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|491
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545
|'''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
|655
|16/11, 22/15
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|764
|[14/9], 11/7, '''25/16'''
|^G, Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|873
|5/3, 18/11, 28/17
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|982
|'''7/4''', 16/9, 30/17
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1091
|['''15/8'''], 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|10
|Octave
|}

== 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)

[[Regular temperament]]s associated with these are discussed in the section below. 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.}}

=== 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. The temperament associated with this equivalence is '''[[Porcupine]]''', where 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. 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]].

== Compositional theory ==
'''PENDING REORGANIZATION FOLLOWING CONSENSUS OF XENGROVE'''

=== Tertian structure ===
22edo has four clear qualities of "thirds" that can serve as mediants in a chord bounded by a fifth. These are the subminor (273{{c}}, 5\22), nearminor (327{{c}}, 6\22), nearmajor (382{{c}}, 7\22), and supermajor (436{{c}}, 8\22) thirds, which reflect the intervals 7/6, 6/5, 5/4, and 9/7 respectively. As the gap between 6/5 and 5/4 is the same as that between 7/6 and 6/5 (or 5/4 and 9/7), 22edo's tertian structure is [[875/864|keemic]].

{| 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. 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.

==== Pythagorean diatonic ====
This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of 22edo. This is good for septimal harmony, not so much for 5-limit harmony (to get 5-limit thirds, you have to go 9 steps up!). A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone (in fact, it is enharmonic to the small whole-tone used in the Zarlino system). This can make interval classifications somewhat unintuitive if ups and downs are not used - for instance, a Nearmajor triad, 0-7-13, is C-D#-G (meaning that a third-sized interval is technically a second) - and in fact 22edo is the first edo other than 15edo (which is strange in its own right) to require ups and downs to notate all intervals without problems with intuition like this (Edos like 17 and 10 utilize semisharps and semiflats, which 22edo cannot use as it does not have neutral intervals.)

==== Zarlino diatonic ====
A faithful way of representing the 5-limit diatonic structure in 22edo (which, unlike its just counterpart, keeps the 7-limit convenient to access) is to use 5/4 as the scale's major third, and likewise 5/3 and 15/8 as the major sixth and seventh respectively. The scale pattern for zarlino is 4-3-2-4-3-4-2. 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), however it does have one thing going for it: in 22edo in particular (and a family of edos including 15, 29, and 37), the interval which separates Pythagorean intervals from their 5-limit counterparts is the exact same as the interval separating 5-limit major and minor intervals. This means that, if # and b are used to represent this interval as an accidental, no additional accidentals are necessary. This is why the 3-step interval is a "Nearmajor second" along with being a chromatic semitone.

==== Blackdye ====
Blackdye is a rank-3 scale similar to zarlino diatonic, which attempts to compromise between zarlino and Pythagorean diatonic in a way, by including a few intervals from both at once. Blackdye can be thought of as dividing each 4-step whole tone into a 3-step tone and a single step called an [[aberrisma]], which separates zarlino and Pythagorean intervals. The blackdye scale pattern is 3-1-3-2-3-1-3-1-3-2. One way to use blackdye is to essentially treat it as multiple overlapping diatonics, which one can modulate between.

==== Pajara ====
Note that the 1-step interval which serves as the 3-limit diatonic semitone is the very same as the 5-limit chromatic semitone, and that the 3-step interval serving as the 3-limit chromatic semitone also happens to map to a certain kind of large 5-limit diatonic semitone (such that it and the chromatic semitone stack to the 4-step whole tone). This suggests that if there was a way to swap the diatonic and chromatic semitones, we could faithfully represent the full 7-limit, including 5.

And as it turns out, there is a way to do that. We start with the pentatonic scale, not only because it's closer to even but because the interval between its small and large steps is exactly the 1-step interval we want to function as our chroma. Then, we add another pentatonic scale that is offset by a tritone from the first one. This results in the decatonic "Pajara[10]" scale (3-2-2-2-2-3-2-2-2-2), where every note has a corresponding note a tritone apart. It solves the problem of representing intervals of both 5 and 7 by introducing three new ordinal classes to provide space for 7-limit intervals to fit on their own degrees of the scale. That way, 7/4 isn't a subminor seventh, it's a major version of the Pajara 8-step. One can even define a notation system for Pajara, wherein the notes are numbered 0-9 and # and b represent alterations by a single step. Pajara retains the property where most notes have a fifth over them.

To extend to the 11-limit, Pajara[12] (1-2-2-2-2-2-1-2-2-2-2-2) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step.

== Tables ==
'''PENDING REORGANIZATION FOLLOWING CONSENSUS OF XENGROVE'''

=== Pajara ===
==== Symmetric scale ====
One possible scale of 22edo, as mentioned previously, is the Pajara[10] decatonic scale, represented as {{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}. This scale can be explored [https://sevish.com/scaleworkshop/?n=22EDO%20jaric%20ssLssssLss%204%7C4%282%29%20soft&l=2Bm_4Bm_7Bm_9Bm_bBm_dBm_fBm_iBm_kBm_mBm&c=&w=r&a=i&y=pi&s=0&r=2o&b=hs&g=gh&version=2.5.7 here]. Below is a chart of its five modes, ordered by rotation. Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
!Mode on fifth
!Mode on fourth
|-
|Dynamic minor
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|
|
|-
|Static minor
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|major
|
|
|-
|Static major
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|
|
|-
|Dynamic major
|{{Interval ruler|22|0, 100, 250, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|
|
|-
|Augmented
|{{Interval ruler|22|0, 150, 250, 400, 500, 600, 750, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
|
|
|}

==== Pentachordal scale ====
This scale is constructed from two identical "pentachords" and the semioctave, and is represented as {{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}. Unlike the symmetric scale, there are ten distinct modes of the pentachordal scale. Ordered by rotation, they are as follows:
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
!Mode on fifth
!Mode on fourth
|-
|Bediyic
|{{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|Hininic
| -
|-
|Alternate minor (Skoronic)
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 950, 1100, 1200}}
|minor
|minor
|dim
|perfect
|minor
|major
|Aujalic
| -
|-
|Moriolic
|{{Interval ruler|22|0, 100, 200, 320, 500, 600, 700, 830, 990, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|major
|major
|Mielauic
|Hininic
|-
|Standard major (Staimosic)
|{{Interval ruler|22|0, 100, 200, 370, 500, 600, 700, 870, 990, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|Prathuic
|Aujalic
|-
|Sebaic
|{{Interval ruler|22|0, 100, 270, 370, 500, 600, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
| -
|Mielauic
|-
|Awanic
|{{Interval ruler|22|0, 170, 270, 370, 500, 670, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
| -
|Prathuic
|-
|Standard minor (Hininic)
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|minor
|Moriolic
|Bediyic
|-
|Aujalic
|{{Interval ruler|22|0, 100, 200, 360, 500, 600, 700, 800, 920, 1110, 1200}}
|minor
|major
|perfect
|perfect
|minor
|major
|Staimosic
|Skoronic
|-
|Alternate major (Kielauic)
|{{Interval ruler|22|0, 100, 270, 360, 500, 600, 700, 800, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|Sebaic
|Moriolic
|-
|Prathuic
|{{Interval ruler|22|0, 170, 270, 360, 500, 600, 700, 870, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|Awanic
|Staimosic
|}
Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].

=== Other scales ===
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 170, 330, 500, 700, 870, 1030, 1200}}
|Approximate Greek scale (equable diatonic), basic MOS of Porcupine.
|-
|Gramitonic (4L5s)
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|Basic MOS of Orwell temperament.
|-
|Zarlino diatonic
|{{Interval ruler|22|0, 110, 330, 500, 700, 800, 1030, 1200}}
|Greek scale (intense diatonic). Zarlino rank-3 diatonic.
|-
|Mosdiatonic
|{{Interval ruler|22|0, 50, 270, 500, 700, 750, 970, 1200}}
|Greek scale (Pythagorean or Archytas diatonic). Basic MOS of Superpyth.
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Pentic
|{{Interval ruler|22|0, 270, 500, 700, 970, 1200}}
|Basic MOS of Superpyth
|}

=== Triads bounded by P5 ===
{| class="wikitable"
!Name
!1
!2
!Bounding interval
!Edostep
!Chart
|-
|Sus4 triad
|Perfect 4th
|Supermajor 2nd
|Perfect 5th
|[0 9 13]
|{{Interval ruler|22|0, 500, 700}}
|-
|Supermajor triad
|Supermajor 3rd
|Subminor 3rd
|Perfect 5th
|[0 8 13]
|{{Interval ruler|22|0, 430, 700}}
|-
|Nearmajor triad
|Nearmajor 3rd
|Nearminor 3rd
|Perfect 5th
|[0 7 13]
|{{Interval ruler|22|0, 380, 700}}
|-
|Nearminor triad
|Nearminor 3rd
|Nearmajor 3rd
|Perfect 5th
|[0 6 13]
|{{Interval ruler|22|0, 320, 700}}
|-
|Subminor triad
|Subminor 3rd
|Supermajor 3rd
|Perfect 5th
|[0 5 13]
|{{Interval ruler|22|0, 270, 700}}
|-
|Sus2 triad
|Supermajor 2nd
|Perfect 4th
|Perfect 5th
|[0 4 13]
|{{Interval ruler|22|0, 200, 700}}
|}

=== Tetrads with P5th ===

==== Harmonic tetrads ====
These represent the conventional structure of a 4:5:6:7 harmonic tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect median above the middle note. This is the characteristic kind of chord found in Pajara temperament, and can also be thought of the stacking of a fifth-bounded triad and a fourth-bounded triad.
{| class="wikitable"
!Name
!1
!2
!3
!4
!Bounding interval 1
!Bounding interval 2
!Bounding interval 3
!Edostep
!Chart
|-
|Nearmajor harmonic tetrad
|Nearmajor 3rd
|Nearminor 3rd
|Subminor 3rd
|Supermajor 2nd
|Perfect 5th
|Subminor 7th
|Perfect 8ve
|[0 7 13 18]
|{{Interval ruler|22|0, 380, 700, 980, 1200}}
|-
|Nearminor harmonic tetrad
|Nearminor 3rd
|Nearmajor 3rd
|Supermajor 2nd
|Subminor 3rd
|Perfect 5th
|Supermajor 6th
|Perfect 8ve
|[0 6 13 17]
|{{Interval ruler|22|0, 320, 700, 920, 1200}}
|}

==== Diatonic tetrads ====
These represent the conventional structure of an 8:10:12:15 diatonic major tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect fifth above the middle note. The sus4 diatonic tetrad is the same as the triad version up to inversion.
{| class="wikitable"
!Name
!1
!2
!3
!Bounding interval 1
!Bounding interval 2
!Edostep
!Chart
|-
|Supermajor diatonic tetrad
|Supermajor 3rd
|Subminor 3rd
|Supermajor 3rd
|Perfect 5th
|Supermajor 7th
|[0 8 13 21]
|{{Interval ruler|22|0, 430, 700, 1150, 1200}}
|-
|Nearmajor diatonic tetrad
|Nearmajor 3rd
|Nearminor 3rd
|Nearmajor 3rd
|Perfect 5th
|Nearmajor 7th
|[0 7 13 20]
|{{Interval ruler|22|0, 380, 700, 1080, 1200}}
|-
|Nearminor diatonic tetrad
|Nearminor 3rd
|Nearmajor 3rd
|Nearminor 3rd
|Perfect 5th
|Nearminor 7th
|[0 6 13 19]
|{{Interval ruler|22|0, 320, 700, 1020, 1200}}
|-
|Subminor diatonic tetrad
|Subminor 3rd
|Supermajor 3rd
|Subminor 3rd
|Perfect 5th
|Subminor 7th
|[0 5 13 18]
|{{Interval ruler|22|0, 270, 700, 970, 1200}}
|-
|Sus2 diatonic tetrad
|Supermajor 2nd
|Perfect 4th
|Supermajor 2nd
|Perfect 5th
|Supermajor 6th
|[0 4 13 17]
|{{Interval ruler|22|0, 210, 700, 920, 1200}}
|}

=== Other triads ===
22edo contains the essentially tempered chord [0 8 16], where the intervals from 1 to 2 and from 2 to 3 are ~9/7, and the interval from 1 to 3 is ~5/3.

{{Cat|Edos}}

MOS

2026-02-13T20:39:50Z

Hkm:

[[File:MOS.png|thumb|401x401px|The comparison of non-MOS and MOS diatonic scales.]]
A '''MOS''' (or '''mos''', or '''moment of symmetry''' scale) is a member of a family of scales which generalize properties of the diatonic and pentatonic scales within 12 equal temperament. Microtonal MOS scales are often seen as some of the easier microtonal scales to work with, because the number of distinct intervals within the scale is quite low.

As in any scale, we can define "n-step interval" to refer to one of many intervals that arise from ascending by n steps of the scale. Formally, a MOS is a scale where, for all n, there are at most two distinct sizes of n-step intervals. Any multiple of the period (which is usually an octave or a fraction thereof) has only one size.[https://en.xen.wiki/w/MOS_scale] MOS scales are also known as '''MV2''' (or '''maximum variety 2''') scales.

MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.

MOSses can be named according to their number of large and small steps (for example, 5L 2s for the diatonic MOS), because there is exactly one step pattern that fits the MOS criteria with any given number of small and large steps. This form of name does not specify the tuning of the MOS or which scale degree is defined as the tonic.

== Examples ==
The most widely used MOS scale is the 12edo [[diatonic]] scale, which has five equal large steps (major seconds) and two equal small steps (minor seconds) within the octave. It can thus be notated 5L 2s. In contrast, while the melodic minor scale (LsLLLLs) has only two step sizes, it is still not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.

A MOS exists for any whole number of large and small steps, for example [[Mosh|3L 4s]] (mosh), which functions as a "neutral" version of the diatonic scale, and [[Onyx|1L 6s]] (onyx), which has 1 large step and thus a very wide range of tunings.

The [[equave]] of a MOS is denoted using angle brackets: for example, 3L2s{{angbr|3/2}} denotes the 3L 2s MOS pattern but using 3/2 as the interval of equivalence rather than 2/1.

== Periods and generators ==
Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] a number of times, then moving each note by multiples of the period (which is usually the octave) to fit within the span of one period.[https://www.anaphoria.com/wilsonintroMOS.html] The latter step is called period reduction. For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, [[Pentic|2L 3s]], is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as C D E F G A C does not produce a MOS, because there are more than 2 sizes of some interval classes.

The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53... However for a quarter-comma [[meantone]] fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50...

A 2/1-equivalent MOS scale aLbs always has 1\gcd(a, b) as its period. For example, 5L2s has period 1\1; 5L5s has period 1\5; 2L8s has period 1\2.

== Hardness ==
One way to specify the tuning of a given MOS pattern (with a given equave) is ''hardness'', which refers to the logarithmic ratio between the size of the L step versus the size of the s step. A tuning of a given MOS that has a higher hardness is ''harder'', and one with a lower hardness is ''softer''.

The hardness of a MOS scale is mostly associated with its melodic shape. Softer tunings (with more similar step sizes) may sound melodically smoother, softer, or more mellow. In contrast, harder tunings of the same MOS (with steps of very different sizes) may sound jagged, dramatic, or sparkly.

Hardness is usually given as L/s, and can fall anywhere between 1 and positive infinity. A hardness value of 1 is called ''equalized'' (since L = s), and positive infinity is called ''collapsed'' (since s = 0). We call hardness 2/1 the ''basic'' tuning of the MOS; the basic tuning is the smallest equal tuning that meaningfully supports the MOS scale. More generally, the basic tuning of a MOS aLbs is always (2a+b)-edo.[https://sevish.com/2021/getting-hard-with-scales/]

Examples for MOS diatonic:
* 12edo diatonic is 2221221, so it has hardness 2/1.
* 17edo diatonic is 3331331, so it has hardness 3/1.
* 19edo diatonic is 3332332, so it has hardness 3/2.
* The equalized tuning is 7edo (1111111).
* The collapsed tuning is 5edo (1110110).

== Table of MOS scales ==
The following table lists common MOS scales grouped by scale size, along with their step patterns and associated [[Temperament|temperaments]]. All temperaments mentioned in the table be found in the [[List of regular temperaments]]. All names in the first column do not specify anything about tuning or which scale degree is the tonic. Common MOSes are highlighted.
{| class="wikitable"
|+ Table of some 2/1-equivalent MOSes.
|-
!Name
!aLbs
!Brightest mode
!Equalized (softest) gen.
!Collapsed (hardest) gen.
!Period (1\1 (1200c) unless otherwise stated)
!Description
|-
! colspan="7" |5-note MOSses
|-
! class="thl" |pentic
! class="thl" |2L3s
|LsLss
|2\5 (480c)
|1\2 (600c)
|
|Called "pentatonic" in 12edo music theory. Five-note subset of both MOS diatonic and antidiatonic.
|-
!|antipentic
!|3L2s
|LLsLs
|2\5 (480c)
|1\3 (400c)
|
|Five-note subset of both oneirotonic and checkertonic.
|-
!|manual
!|4L1s
|LLLLs
|1\5 (240c)
|1\4 (300c)
|
|Five-note subset of both semiquartal and gramitonic.
|-
! colspan="7" |6-note MOSses
|-
!|machinoid
!|5L1s
|LLLLLs
|1\6 (200c)
|1\5 (240c)
|
|Some temperament interpretations: Machine[6], Gorgo[6], Slendric[6].
|-
! colspan="7" |7-note MOSses
|-
! class="thl" |onyx
! class="thl" |1L6s
|Lssssss
|1\7 (171.4c)
|0\6 (0c)
|
|One temperament interpretation is Porcupine[7].
|-
!|antidiatonic
!|2L5s
|LssLsss
|3\7 (685.7c)
|1\2 (600c)
|
|When using a very flat fifth, reverses the interval qualities of diatonic. One temperament interpretation is Mabilic[7].
|-
! class="thl" |mosh
! class="thl" |3L4s
|LsLsLss
|2\7 (342.9c)
|1\3 (400c)
|
|Neutral thirds generate this MOS.
|-
!|smitonic
!|4L3s
|LLsLsLs
|2\7 (342.9c)
|1\4 (300c)
|
|So named because the generator is a sharp minor third. Inthar finds that it sounds like a brighter, stretched version of the diatonic scale. In fact, it can be seen as a "warped" diatonic scale, because it can be found by replacing one large step of diatonic with a small step. One temperament interpretation is Orgone[7].
|-
! class="thl" |(MOS) diatonic
! class="thl" |5L2s
|LLLsLLs
|4\7 (685.7c)
|3\5 (720c)
|
|
|-
!|arch(a)eotonic
!|6L1s
|LLLLLLs
|1\7 (171.4c)
|1\6 (200c)
|
|{{Adv|Some temperament interpretations: Tetracot[7], Didacus[7].}}
|-
! colspan="7" |8-note MOSses
|-
!|checkertonic
!|3L5s
|LsLssLss
|3\8 (450c)
|1\3 (400c)
|
|Somewhat like a stretched tcherepnin scale. Associated with Squares (when relatively hard) and Sensi (hardness 3/2 or softer). In softer tunings, it is the most consonant 8-tone MOS.
|-
! class="thl" |tetrawood
! class="thl" |4L4s
|LsLsLsLs
|1\8 (150c)
|1\4 (300c)
|1\4 (300c)
|Exists in 12edo; often called the "diminished" or "octatonic" scale in 12edo theory.[https://en.wikipedia.org/wiki/Octatonic_scale]
|-
!|oneirotonic
!|5L3s
|LLsLLsLs
|3\8 (450c)
|2\5 (480c)
|
|Sounds like a darker, compressed version of the diatonic scale according to Inthar.
|-
!|ekic
!|6L2s
|LLLsLLLs
|1\8 (150c)
|1\6 (200c)
|1\2 (600c)
| Adv |The 22edo (step ratio L:s = 3:2) tuning is associated with Hedgehog temperament. Lumi Pakkanen has described the sound of this MOS as "dreamy, yet oppressing", especially when tuned to step ratio 3:1.[https://en.xen.wiki/w/6L_2s]
|-
! class="thl" |pine
! class="thl" |7L1s
|LLLLLLLs
|1\8 (150c)
|1\7 (171.4c)
|
|
|-
! colspan="7" |9-note MOSses
|-
!|tcherepnin
!|3L6s
|LssLssLss
|1\9 (133.3c)
|1\3 (400c)
|1\3 (400c)
|Exists in 12edo; associated with Augmented temperament, which collapses the circle of just (5/4) major thirds into one augmented chord formed from 400c major thirds.
|-
!|gramitonic
!|4L5s
|LsLsLsLss
|2\9 (266.7c)
|1\4 (300c)
|
|So named because the generator is a grave minor third. Associated with Orwell temperament. One of the more consonant 9-note MOSses, especially when tuned with a step ratio 4:3 < L:s < 3:2.
|-
! class="thl" |semiquartal
! class="thl" |5L4s
|LLsLsLsLs
|2\9 (266.7c)
|1\5 (240c)
|
|So named because the generator is half a fourth. One of the more consonant 9-note MOSses.
|-
!|armotonic
!|7L2s
|LLLLsLLLs
|5\9 (666.7c)
|4\7 (685.7c)
|
|Contains antidiatonic as a subset. One temperament interpretation is Mabilic[9], or it can be seen in lower complexity as Mavila[9], where three just perfect fourths stack to a just (5/2) major tenth.
|-
!|subneutralic
!|8L1s
|LLLLLLLLs
|1\9 (133.3c)
|1\8 (150c)
|
|Has tunings that split the perfect fifth into 5 equal parts, e.g. in 17edo.
|-
! colspan="7" |10-note MOSses
|-
!|jaric
!|2L8s
|LssssLssss
|1\10 (120c)
|1\2 (600c)
|1\2 (600c)
|{{Adv|So named because of Pajara[10] and Injera[10] interpretations. Interpreted as Diaschismic[10] in 34edo and 46edo.}}
|-
! class="thl" |pentawood
! class="thl" |5L5s
|LsLsLsLsLs
|1\10 (120c)
|0\5 (0c)
|1\5 (240c)
|One temperament interpretation is Blackwood[10]. Often called the Blackwood scale because composer Easley Blackwood was one of the first to use it. Contains either a major or minor triad on every note, and thus can be seen as having similar properties to MOSdiatonic in 5n-edos.
|-
!|dicoid
!|7L3s
|LLLsLLsLLs
|3\10 (360c)
|2\7 (342.9c)
|
|The 10-note MOS generated by neutral thirds. So named because of the exotemperament Dichotic.
|-
!|taric
!|8L2s
|LLLLsLLLLs
|1\10 (120c)
|1\8 (150c)
|1\2 (600c)
|Generated by an oneirotonic generator (3\5 to 5\8). Named after Hindi for 18 (''aṭhārah''), because 18edo is the basic tuning.
|-
!|sinatonic
!|9L1s
|LLLLLLLLLs
|1\10 (120c)
|1\9 (133.3c)
|
|So named because of the "sinaic" generator (named after ibn Sina), which is 1/4 of a perfect fourth.
|-
! colspan="7" |Larger MOSses
|-
!|slentonic
!|5L6s
|sLsLsLsLsLs
|2\11 (218.2c)
|1\5 (240c)
|
|The 11-note MOS of Slendric.
|-
!|p-chro smitonic
!|4L7s
|LsLssLssLss
|3\11 (327.3c)
|1\4 (300c)
|
|The 11-note MOS associated with Kleismic/Cata and Orgone, both accurate temperaments. Older material may call this scale "kleistonic."
|-
! class="thl" |p-chromatic
! class="thl" |5L7s
|LsLsLssLsLss
|7\12 (700c)
|3\5 (720c)
|
|The chromatic scale generated by sharp-of-12edo fifths. Superpyth[12] is a particularly interesting interpretation.
|-
!|hexawood
!|6L6s
|LsLsLsLsLsLs
|1\12 (100c)
|1\6 (200c)
|1\6 (200c)
|A "straddle-fifth chromatic scale", as it can be constructed by stacking alternating flat and sharp fifths as long as they stack to 1\6, in e.g. 18edo.
|-
! class="thl" |m-chromatic
! class="thl" |7L5s
|LLsLsLLsLsLs
|7\12 (700c)
|4\7 (685.7c)
|
|The chromatic scale generated by flat-of-12edo fifths. Used in many 17th-century keyboards and still used in some church organs.
|-
!|telluric
!|10L2s
|LLLLLsLLLLLs
|1\12 (100c)
|1\10 (120c)
|1\2 (600c)
|Commonly interpreted as Pajara[12] or Diaschismic[12].
|-
!|heptawood
!|7L7s
|LsLsLsLsLsLsLs
|1\14 (85.7c)
|0\7 (0c)
|1\7 (171.4c)
|Two offset rings of 7edo fifths; the 7edo analogue of the blackwood MOS.
|-
!|
!|11L4s
|LLLsLLLsLLLsLLs
|4\15 (320c)
|3\11 (327.3c)
|
|The first MOS unambiguously interpreted as Orgone.
|-
!|
!|15L4s
|LLLLsLLLLsLLLLsLLLs
|7\19 (315.8c)
|4\15 (320c)
|
|The first MOS unambiguously interpreted as Kleismic.
|}

{{Cat|
Scale construction
Core knowledge
}}

User:Hkm

2026-02-13T02:35:19Z

Hkm: Blanked the page

MOS

2026-02-12T07:01:58Z

Hkm: not done fixing this page. i hope to do some more work on it later

[[File:MOS.png|thumb|401x401px|The comparison of non-MOS and MOS diatonic scales.]]
A '''MOS''' (or '''mos''', or '''moment of symmetry''' scale) is a member of a family of scales which generalize properties of the diatonic and pentatonic scales within 12 equal temperament. Microtonal MOS scales are often seen as some of the easier microtonal scales to work with, because the umber of distinct intervals within the scale is quite low.

Formally, a MOS is a scale where every step is either small or large (with no in-between), and the same is true with any interval formed by two adjacent steps (a "2-step"), by three consecutive steps (a "3-step"), etc. Any multiple of the period (which is usually an octave or a fraction thereof) has only one size.[https://en.xen.wiki/w/MOS_scale] MOS scales are also known as '''MV2''' (or '''maximum variety 2''') scales.

MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.

== Examples ==
The most widely used MOS scale is the MOS form of the [[diatonic]] scale, which has five equal large steps (major seconds) and two equal small steps (minor seconds) within the octave. It can thus be notated 5L 2s, and it can be shown that there is a unique scale (counting [[Mode|rotation]]s as the same scale) that meets the MOS criteria with a given number of large and small steps. For example, the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.

A MOS exists for any whole number of large and small steps, for example [[Mosh|3L 4s]] (mosh), which functions as a "neutral" version of the diatonic scale, and [[Onyx|1L 6s]] (onyx), which has 1 large step and thus a very wide range of tunings.

The [[equave]] of a MOS is denoted using angle brackets: for example, 3L2s{{angbr|3/2}} denotes the 3L 2s MOS pattern but using 3/2 as the interval of equivalence rather than 2/1.

== Periods and generators ==
Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and [[octave-reducing]] (or more generally, [[Generator|period]]-reducing).[https://www.anaphoria.com/wilsonintroMOS.html] For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, [[Pentic|2L 3s]], is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as C D E F G A C does not produce a MOS, because there are more than 2 sizes of each interval class.

The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53... However for a quarter-comma [[meantone]] fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50...

A 2/1-equivalent MOS scale aLbs always has 1\gcd(a, b) as its period. For example, 5L2s has period 1\1; 5L5s has period 1\5; 2L8s has period 1\2.

== Hardness ==
One way to specify the tuning of a given MOS pattern (with a given equave) is ''hardness'', which refers to the logarithmic ratio between the size of the L step versus the size of the s step. A tuning of a given MOS that has a higher hardness is ''harder'', and one with a lower hardness is ''softer''.

The hardness of a MOS scale is mostly associated with its melodic shape. Softer tunings (with more similar step sizes) may sound melodically smoother, softer, or more mellow. In contrast, harder tunings of the same MOS (with steps of very different sizes) may sound jagged, dramatic, or sparkly.

Hardness is usually given as L/s, and can fall anywhere between 1 and positive infinity. A hardness value of 1 is called ''equalized'' (since L = s), and positive infinity is called ''collapsed'' (since s = 0). We call hardness 2/1 the ''basic'' tuning of the MOS; the basic tuning is the smallest equal tuning that meaningfully supports the MOS scale. More generally, the basic tuning of a MOS aLbs is always (2a+b)-edo.[https://sevish.com/2021/getting-hard-with-scales/]

Examples for MOS diatonic:
* 12edo diatonic is 2221221, so it has hardness 2/1.
* 17edo diatonic is 3331331, so it has hardness 3/1.
* 19edo diatonic is 3332332, so it has hardness 3/2.
* The equalized tuning is 7edo (1111111).
* The collapsed tuning is 5edo (1110110).

== Table of MOS scales ==
{| class="wikitable"
|+ Table of some 2/1-equivalent MOSes. Temperaments are capitalized and can be found in the [[List of regular temperaments]]. Common MOSes are highlighted.
|-
!Name
!aLbs
!Brightest mode
!Equalized (softest) gen.
!Collapsed (hardest) gen.
!Period (1\1 (1200c) unless otherwise stated)
!Description
|-
! colspan="7" |5-note MOSses
|-
!class="thl"|pentic
!class="thl"|2L3s
|LsLss
|2\5 (480c)
|1\2 (600c)
|
|Called "pentatonic" in 12edo music theory. Five-note subset of both MOS diatonic and antidiatonic.
|-
!|antipentic
!|3L2s
|LLsLs
|2\5 (480c)
|1\3 (400c)
|
|Five-note subset of both oneirotonic and checkertonic.
|-
!|manual
!|4L1s
|LLLLs
|1\5 (240c)
|1\4 (300c)
|
|Five-note subset of both semiquartal and gramitonic.
|-
! colspan="7" |6-note MOSses
|-
!|machinoid
!|5L1s
|LLLLLs
|1\6 (200c)
|1\5 (240c)
|
|Some temperament interpretations: Machine[6], Gorgo[6], Slendric[6].
|-
! colspan="7" |7-note MOSses
|-
!class="thl"|onyx
!class="thl"|1L6s
|Lssssss
|1\7 (171.4c)
|0\6 (0c)
|
|One temperament interpretation is Porcupine[7].
|-
!|antidiatonic
!|2L5s
|LssLsss
|3\7 (685.7c)
|1\2 (600c)
|
|When using a very flat fifth, reverses the interval qualities of diatonic. One temperament interpretation is Mabilic[7].
|-
!class="thl"|mosh
!class="thl"|3L4s
|LsLsLss
|2\7 (342.9c)
|1\3 (400c)
|
|Neutral thirds generate this MOS.
|-
!|smitonic
!|4L3s
|LLsLsLs
|2\7 (342.9c)
|1\4 (300c)
|
|So named because the generator is a sharp minor third. Inthar finds that it sounds like a brighter, stretched version of the diatonic scale. In fact, it can be seen as a "warped" diatonic scale, because it can be found by replacing one large step of diatonic with a small step. One temperament interpretation is Orgone[7].
|-
!class="thl"|(MOS) diatonic
!class="thl"|5L2s
|LLLsLLs
|4\7 (685.7c)
|3\5 (720c)
|
|
|-
!|arch(a)eotonic
!|6L1s
|LLLLLLs
|1\7 (171.4c)
|1\6 (200c)
|
|{{Adv|Some temperament interpretations: Tetracot[7], Didacus[7].}}
|-
! colspan="7" |8-note MOSses
|-
!|checkertonic
!|3L5s
|LsLssLss
|3\8 (450c)
|1\3 (400c)
|
|Somewhat like a stretched tcherepnin scale. Associated with Squares (when relatively hard) and Sensi (hardness 3/2 or softer). In softer tunings, it is the most consonant 8-tone MOS.
|-
! class="thl" |tetrawood
! class="thl" |4L4s
|LsLsLsLs
|1\8 (150c)
|1\4 (300c)
|1\4 (300c)
|Exists in 12edo; often called the "diminished" or "octatonic" scale in 12edo theory.[https://en.wikipedia.org/wiki/Octatonic_scale]
|-
!|oneirotonic
!|5L3s
|LLsLLsLs
|3\8 (450c)
|2\5 (480c)
|
|Sounds like a darker, compressed version of the diatonic scale according to Inthar.
|-
!|ekic
!|6L2s
|LLLsLLLs
|1\8 (150c)
|1\6 (200c)
|1\2 (600c)
|Adv|The 22edo (step ratio L:s = 3:2) tuning is associated with Hedgehog temperament. Lumi Pakkanen has described the sound of this MOS as "dreamy, yet oppressing", especially when tuned to step ratio 3:1.[https://en.xen.wiki/w/6L_2s]
|-
!class="thl"|pine
!class="thl"|7L1s
|LLLLLLLs
|1\8 (150c)
|1\7 (171.4c)
|
|
|-
! colspan="7" |9-note MOSses
|-
!|tcherepnin
!|3L6s
|LssLssLss
|1\9 (133.3c)
|1\3 (400c)
|1\3 (400c)
|Exists in 12edo; associated with Augmented temperament, which collapses the circle of just (5/4) major thirds into one augmented chord formed from 400c major thirds.
|-
!|gramitonic
!|4L5s
|LsLsLsLss
|2\9 (266.7c)
|1\4 (300c)
|
|So named because the generator is a grave minor third. Associated with Orwell temperament. One of the more consonant 9-note MOSses, especially when tuned with a step ratio 4:3 < L:s < 3:2.
|-
!class="thl"|semiquartal
!class="thl"|5L4s
|LLsLsLsLs
|2\9 (266.7c)
|1\5 (240c)
|
|So named because the generator is half a fourth. One of the more consonant 9-note MOSses.
|-
!|armotonic
!|7L2s
|LLLLsLLLs
|5\9 (666.7c)
|4\7 (685.7c)
|
|Contains antidiatonic as a subset. One temperament interpretation is Mabilic[9], or it can be seen in lower complexity as Mavila[9], where three just perfect fourths stack to a just (5/2) major tenth.
|-
!|subneutralic
!|8L1s
|LLLLLLLLs
|1\9 (133.3c)
|1\8 (150c)
|
|Has tunings that split the perfect fifth into 5 equal parts, e.g. in 17edo.
|-
! colspan="7" |10-note MOSses
|-
!|jaric
!|2L8s
|LssssLssss
|1\10 (120c)
|1\2 (600c)
|1\2 (600c)
|{{Adv|So named because of Pajara[10] and Injera[10] interpretations. Interpreted as Diaschismic[10] in 34edo and 46edo.}}
|-
!class="thl"|pentawood
!class="thl"|5L5s
|LsLsLsLsLs
|1\10 (120c)
|0\5 (0c)
|1\5 (240c)
|One temperament interpretation is Blackwood[10]. Often called the Blackwood scale because composer Easley Blackwood was one of the first to use it. Contains either a major or minor triad on every note, and thus can be seen as having similar properties to MOSdiatonic in 5n-edos.
|-
!|dicoid
!|7L3s
|LLLsLLsLLs
|3\10 (360c)
|2\7 (342.9c)
|
|The 10-note MOS generated by neutral thirds. So named because of the exotemperament Dichotic.
|-
!|taric
!|8L2s
|LLLLsLLLLs
|1\10 (120c)
|1\8 (150c)
|1\2 (600c)
|Generated by an oneirotonic generator (3\5 to 5\8). Named after Hindi for 18 (''aṭhārah''), because 18edo is the basic tuning.
|-
!|sinatonic
!|9L1s
|LLLLLLLLLs
|1\10 (120c)
|1\9 (133.3c)
|
|So named because of the "sinaic" generator (named after ibn Sina), which is 1/4 of a perfect fourth.
|-
! colspan="7" |Larger MOSses
|-
!|slentonic
!|5L6s
|sLsLsLsLsLs
|2\11 (218.2c)
|1\5 (240c)
|
|The 11-note MOS of Slendric.
|-
!|p-chro smitonic
!|4L7s
|LsLssLssLss
|3\11 (327.3c)
|1\4 (300c)
|
|The 11-note MOS associated with Kleismic/Cata and Orgone, both accurate temperaments. Older material may call this scale "kleistonic."
|-
!class="thl"|p-chromatic
!class="thl"|5L7s
|LsLsLssLsLss
|7\12 (700c)
|3\5 (720c)
|
|The chromatic scale generated by sharp-of-12edo fifths. Superpyth[12] is a particularly interesting interpretation.
|-
!|hexawood
!|6L6s
|LsLsLsLsLsLs
|1\12 (100c)
|1\6 (200c)
|1\6 (200c)
|A "straddle-fifth chromatic scale", as it can be constructed by stacking alternating flat and sharp fifths as long as they stack to 1\6, in e.g. 18edo.
|-
!class="thl"|m-chromatic
!class="thl"|7L5s
|LLsLsLLsLsLs
|7\12 (700c)
|4\7 (685.7c)
|
|The chromatic scale generated by flat-of-12edo fifths. Used in many 17th-century keyboards and still used in some church organs.
|-
!|telluric
!|10L2s
|LLLLLsLLLLLs
|1\12 (100c)
|1\10 (120c)
|1\2 (600c)
|Commonly interpreted as Pajara[12] or Diaschismic[12].
|-
!|heptawood
!|7L7s
|LsLsLsLsLsLsLs
|1\14 (85.7c)
|0\7 (0c)
|1\7 (171.4c)
|Two offset rings of 7edo fifths; the 7edo analogue of the blackwood MOS.
|-
!|
!|11L4s
|LLLsLLLsLLLsLLs
|4\15 (320c)
|3\11 (327.3c)
|
|The first MOS unambiguously interpreted as Orgone.
|-
!|
!|15L4s
|LLLLsLLLLsLLLLsLLLs
|7\19 (315.8c)
|4\15 (320c)
|
|The first MOS unambiguously interpreted as Kleismic.
|}

{{Cat|
Scale construction
Core knowledge
}}

22edo

2026-02-12T06:34:03Z

Hkm: /* Triads and tetrads */ liberating the compositionally useful bits from the rtt framework :trollet:

'''22edo''', or 22 equal divisions of the octave, 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 [[9/8]] and [[8/7]] are the same interval. 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.

22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]], which ADIN calls "nearminor" and "nearmajor" thirds. As a result, 22 is perhaps the smallest EDO that can be considered to include 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.

== General theory ==
=== 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)

=== 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 of an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 16/9 with 7/4
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[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)

[[Regular temperament]]s associated with these are discussed in [[#Temperaments and generators]]. 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.}}

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the 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.

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.

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 and notation ===
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).

The "native-fifths" system mentioned here is the most commonly used system, and the one that most microtonal notation systems support by default. It is derived through stacking 22edo's tempered version of 3/2 and assigning names accordingly. As a result, the nominals (C, D, E, F, G, ...) follow the diatonic MOS, where the distances between the large steps (C-D, D-E, F-G, G-A, A-B) are 4 EDO steps, and the distances between the small steps (E-F, B-C) are 1 EDO step. Therefore, a sharp corresponds to +3 EDO steps while a flat corresponds to -3 (representing the diatonic chroma in each case). Each sharp or flat can be split into three distinct notes, so we use the accidental ^ to raise by one EDO step and v to lower by one EDO step.

In the other proposed notation systems aside from native fifths, a sharp corresponds to +1 EDO step, while a flat corresponds to -1. The Zarlino notation here uses the [[ternary]] Zarlino scale (see [[#Zarlino diatonic]]), or Ptolemy's intense diatonic, as its basic scale (which prioritizes the 5-limit, whereas native fifths prioritize 2.3.7), and uses to its advantage the fact that one step of 22edo maps to both 25/24 and 81/80 (a property of [[Porcupine]] temperament). Pajara uses the 10-note Pajara scale (see [[#Pajara]]) as its basis, which is generated by a perfect fifth but splits the octave in two to reach intervals of the full 7-limit relatively easily; the scale has four 2-step and one 3-step intervals per half-octave, which differ in size by a diatonic minor second, which is 1 step in 22edo.

The ADIN system uses the labels "nearminor" and "nearmajor" for intervals that 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. Note also that as "subminor" and "supermajor" intervals are the minor and major intervals of the diatonic MOS in 22edo, unqualified "major" and "minor" by default refer to these.

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 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
|55
|25/24, 28/27, [33/32], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|164
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218
|8/7, '''9/8''', 17/15
|D
|D#
|2
|(Super)major second
|-
|5
|273
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327
|6/5, 11/9
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|382
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|491
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545
|'''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
|655
|16/11
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|764
|[14/9], 11/7, 25/16
|Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|873
|5/3, 18/11
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|982
|'''7/4''', 16/9, 30/17
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1091
|'''[15/8]''', 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|10
|Octave
|}

== Structural theory ==
=== Temperaments and generators ===
==== Whole-octave temperaments ====
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. The temperament associated with this equivalence is '''Porcupine''', where 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.

==== Split-octave temperaments ====
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. 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]].

=== Tertian structure ===
22edo has four clear qualities of "thirds" that can serve as mediants in a chord bounded by a fifth. These are the subminor (273{{c}}, 5\22), nearminor (327{{c}}, 6\22), nearmajor (382{{c}}, 7\22), and supermajor (436{{c}}, 8\22) thirds, which reflect the intervals 7/6, 6/5, 5/4, and 9/7 respectively. As the gap between 6/5 and 5/4 is the same as that between 7/6 and 6/5 (or 5/4 and 9/7), 22edo's tertian structure is [[875/864|keemic]].

{| 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. 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.

=== Pythagorean diatonic ===
This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of 22edo. This is good for septimal harmony, not so much for 5-limit harmony (to get 5-limit thirds, you have to go 9 steps up!). A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone (in fact, it is enharmonic to the small whole-tone used in the Zarlino system). This can make interval classifications somewhat unintuitive if ups and downs are not used - for instance, a Nearmajor triad, 0-7-13, is C-D#-G (meaning that a third-sized interval is technically a second) - and in fact 22edo is the first edo other than 15edo (which is strange in its own right) to require ups and downs to notate all intervals without problems with intuition like this (Edos like 17 and 10 utilize semisharps and semiflats, which 22edo cannot use as it does not have neutral intervals.)

=== Zarlino diatonic ===
A faithful way of representing the 5-limit diatonic structure in 22edo (which, unlike its just counterpart, keeps the 7-limit convenient to access) is to use 5/4 as the scale's major third, and likewise 5/3 and 15/8 as the major sixth and seventh respectively. The scale pattern for zarlino is 4-3-2-4-3-4-2. 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), however it does have one thing going for it: in 22edo in particular (and a family of edos including 15, 29, and 37), the interval which separates Pythagorean intervals from their 5-limit counterparts is the exact same as the interval separating 5-limit major and minor intervals. This means that, if # and b are used to represent this interval as an accidental, no additional accidentals are necessary. This is why the 3-step interval is a "Nearmajor second" along with being a chromatic semitone.

=== Pajara ===
Note that the 1-step interval which serves as the 3-limit diatonic semitone is the very same as the 5-limit chromatic semitone, and that the 3-step interval serving as the 3-limit chromatic semitone also happens to map to a certain kind of large 5-limit diatonic semitone (such that it and the chromatic semitone stack to the 4-step whole tone). This suggests that if there was a way to swap the diatonic and chromatic semitones, we could faithfully represent the full 7-limit, including 5.

And as it turns out, there is a way to do that. We start with the pentatonic scale, not only because it's closer to even but because the interval between its small and large steps is exactly the 1-step interval we want to function as our chroma. Then, we add another pentatonic scale that is offset by a tritone from the first one. This results in the decatonic "Pajara[10]" scale (3-2-2-2-2-3-2-2-2-2), where every note has a corresponding note a tritone apart. It solves the problem of representing intervals of both 5 and 7 by introducing three new ordinal classes to provide space for 7-limit intervals to fit on their own degrees of the scale. That way, 7/4 isn't a subminor seventh, it's a major version of the Pajara 8-step. One can even define a notation system for Pajara, wherein the notes are numbered 0-9 and # and b represent alterations by a single step. Pajara retains the property where most notes have a fifth over them.

To extend to the 11-limit, Pajara[12] (1-2-2-2-2-2-1-2-2-2-2-2) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step.

==== Symmetric scale ====
One possible scale of 22edo, as mentioned previously, is the Pajara[10] decatonic scale, represented as {{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}. This scale can be explored [https://sevish.com/scaleworkshop/?n=22EDO%20jaric%20ssLssssLss%204%7C4%282%29%20soft&l=2Bm_4Bm_7Bm_9Bm_bBm_dBm_fBm_iBm_kBm_mBm&c=&w=r&a=i&y=pi&s=0&r=2o&b=hs&g=gh&version=2.5.7 here]. Below is a chart of its five modes, ordered by rotation. Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
!Mode on fifth
!Mode on fourth
|-
|Dynamic minor
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|
|
|-
|Static minor
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|major
|
|
|-
|Static major
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|
|
|-
|Dynamic major
|{{Interval ruler|22|0, 100, 250, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|
|
|-
|Augmented
|{{Interval ruler|22|0, 150, 250, 400, 500, 600, 750, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
|
|
|}

==== Pentachordal scale ====
This scale is constructed from two identical "pentachords" and the semioctave, and is represented as {{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}. Unlike the symmetric scale, there are ten distinct modes of the pentachordal scale. Ordered by rotation, they are as follows:
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
!Mode on fifth
!Mode on fourth
|-
|Bediyic
|{{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|Hininic
| -
|-
|Alternate minor (Skoronic)
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 950, 1100, 1200}}
|minor
|minor
|dim
|perfect
|minor
|major
|Aujalic
| -
|-
|Moriolic
|{{Interval ruler|22|0, 100, 200, 320, 500, 600, 700, 830, 990, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|major
|major
|Mielauic
|Hininic
|-
|Standard major (Staimosic)
|{{Interval ruler|22|0, 100, 200, 370, 500, 600, 700, 870, 990, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|Prathuic
|Aujalic
|-
|Sebaic
|{{Interval ruler|22|0, 100, 270, 370, 500, 600, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
| -
|Mielauic
|-
|Awanic
|{{Interval ruler|22|0, 170, 270, 370, 500, 670, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
| -
|Prathuic
|-
|Standard minor (Hininic)
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|minor
|Moriolic
|Bediyic
|-
|Aujalic
|{{Interval ruler|22|0, 100, 200, 360, 500, 600, 700, 800, 920, 1110, 1200}}
|minor
|major
|perfect
|perfect
|minor
|major
|Staimosic
|Skoronic
|-
|Alternate major (Kielauic)
|{{Interval ruler|22|0, 100, 270, 360, 500, 600, 700, 800, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|Sebaic
|Moriolic
|-
|Prathuic
|{{Interval ruler|22|0, 170, 270, 360, 500, 600, 700, 870, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|Awanic
|Staimosic
|}
Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].

=== Blackdye ===
Blackdye is a rank-3 scale similar to zarlino diatonic, which attempts to compromise between zarlino and Pythagorean diatonic in a way, by including a few intervals from both at once. Blackdye can be thought of as dividing each 4-step whole tone into a 3-step tone and a single step called an [[aberrisma]], which separates zarlino and Pythagorean intervals. The blackdye scale pattern is 3-1-3-2-3-1-3-1-3-2. One way to use blackdye is to essentially treat it as multiple overlapping diatonics, which one can modulate between.

=== Other scales ===
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 170, 330, 500, 700, 870, 1030, 1200}}
|Approximate Greek scale (equable diatonic), basic MOS of Porcupine.
|-
|Gramitonic (4L5s)
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|Basic MOS of Orwell temperament.
|-
|Zarlino diatonic
|{{Interval ruler|22|0, 110, 330, 500, 700, 800, 1030, 1200}}
|Greek scale (intense diatonic). Zarlino rank-3 diatonic.
|-
|Mosdiatonic
|{{Interval ruler|22|0, 50, 270, 500, 700, 750, 970, 1200}}
|Greek scale (Pythagorean or Archytas diatonic). Basic MOS of Superpyth.
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Pentic
|{{Interval ruler|22|0, 270, 500, 700, 970, 1200}}
|Basic MOS of Superpyth
|}

== Triads and tetrads ==

=== Triads bounded by P5 ===
{| class="wikitable"
!Name
!1
!2
!Bounding interval
!Edostep
!Chart
|-
|Sus4 triad
|Perfect 4th
|Supermajor 2nd
|Perfect 5th
|[0 9 13]
|{{Interval ruler|22|0, 500, 700}}
|-
|Supermajor triad
|Supermajor 3rd
|Subminor 3rd
|Perfect 5th
|[0 8 13]
|{{Interval ruler|22|0, 430, 700}}
|-
|Nearmajor triad
|Nearmajor 3rd
|Nearminor 3rd
|Perfect 5th
|[0 7 13]
|{{Interval ruler|22|0, 380, 700}}
|-
|Nearminor triad
|Nearminor 3rd
|Nearmajor 3rd
|Perfect 5th
|[0 6 13]
|{{Interval ruler|22|0, 320, 700}}
|-
|Subminor triad
|Subminor 3rd
|Supermajor 3rd
|Perfect 5th
|[0 5 13]
|{{Interval ruler|22|0, 270, 700}}
|-
|Sus2 triad
|Supermajor 2nd
|Perfect 4th
|Perfect 5th
|[0 4 13]
|{{Interval ruler|22|0, 200, 700}}
|}

=== Tetrads with P5th ===

==== Harmonic tetrads ====
These represent the conventional structure of a 4:5:6:7 harmonic tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect median above the middle note. This is the characteristic kind of chord found in Pajara temperament, and can also be thought of the stacking of a fifth-bounded triad and a fourth-bounded triad.
{| class="wikitable"
!Name
!1
!2
!3
!4
!Bounding interval 1
!Bounding interval 2
!Bounding interval 3
!Edostep
!Chart
|-
|Nearmajor harmonic tetrad
|Nearmajor 3rd
|Nearminor 3rd
|Subminor 3rd
|Supermajor 2nd
|Perfect 5th
|Subminor 7th
|Perfect 8ve
|[0 7 13 18]
|{{Interval ruler|22|0, 380, 700, 980, 1200}}
|-
|Nearminor harmonic tetrad
|Nearminor 3rd
|Nearmajor 3rd
|Supermajor 2nd
|Subminor 3rd
|Perfect 5th
|Supermajor 6th
|Perfect 8ve
|[0 6 13 17]
|{{Interval ruler|22|0, 320, 700, 920, 1200}}
|}

==== Diatonic tetrads ====
These represent the conventional structure of an 8:10:12:15 diatonic major tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect fifth above the middle note. The sus4 diatonic tetrad is the same as the triad version up to inversion.
{| class="wikitable"
!Name
!1
!2
!3
!Bounding interval 1
!Bounding interval 2
!Edostep
!Chart
|-
|Supermajor diatonic tetrad
|Supermajor 3rd
|Subminor 3rd
|Supermajor 3rd
|Perfect 5th
|Supermajor 7th
|[0 8 13 21]
|{{Interval ruler|22|0, 430, 700, 1150, 1200}}
|-
|Nearmajor diatonic tetrad
|Nearmajor 3rd
|Nearminor 3rd
|Nearmajor 3rd
|Perfect 5th
|Nearmajor 7th
|[0 7 13 20]
|{{Interval ruler|22|0, 380, 700, 1080, 1200}}
|-
|Nearminor diatonic tetrad
|Nearminor 3rd
|Nearmajor 3rd
|Nearminor 3rd
|Perfect 5th
|Nearminor 7th
|[0 6 13 19]
|{{Interval ruler|22|0, 320, 700, 1020, 1200}}
|-
|Subminor diatonic tetrad
|Subminor 3rd
|Supermajor 3rd
|Subminor 3rd
|Perfect 5th
|Subminor 7th
|[0 5 13 18]
|{{Interval ruler|22|0, 270, 700, 970, 1200}}
|-
|Sus2 diatonic tetrad
|Supermajor 2nd
|Perfect 4th
|Supermajor 2nd
|Perfect 5th
|Supermajor 6th
|[0 4 13 17]
|{{Interval ruler|22|0, 210, 700, 920, 1200}}
|}

=== Other triads ===
22edo contains the essentially tempered chord [0 8 16], where the intervals from 1 to 2 and from 2 to 3 are ~9/7, and the interval from 1 to 3 is ~5/3.{{Cat|Edos}}

22edo

2026-02-12T06:29:28Z

Hkm: /* Temperaments and generators */ I think this definitely needs to go; we should also consider removing the sections on Orwell and Magic, because stacking those generators enough to reach interesting equivalences doesn't really happen naturally. Magic's property of 2 ~5/4s making a 11/7~14/9 (which is an interesting equivalence) is already mentioned on the interval table.

'''22edo''', or 22 equal divisions of the octave, 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 [[9/8]] and [[8/7]] are the same interval. 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.

22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]], which ADIN calls "nearminor" and "nearmajor" thirds. As a result, 22 is perhaps the smallest EDO that can be considered to include 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.

== General theory ==
=== 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)

=== 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 of an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 16/9 with 7/4
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[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)

[[Regular temperament]]s associated with these are discussed in [[#Temperaments and generators]]. 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.}}

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the 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.

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.

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 and notation ===
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).

The "native-fifths" system mentioned here is the most commonly used system, and the one that most microtonal notation systems support by default. It is derived through stacking 22edo's tempered version of 3/2 and assigning names accordingly. As a result, the nominals (C, D, E, F, G, ...) follow the diatonic MOS, where the distances between the large steps (C-D, D-E, F-G, G-A, A-B) are 4 EDO steps, and the distances between the small steps (E-F, B-C) are 1 EDO step. Therefore, a sharp corresponds to +3 EDO steps while a flat corresponds to -3 (representing the diatonic chroma in each case). Each sharp or flat can be split into three distinct notes, so we use the accidental ^ to raise by one EDO step and v to lower by one EDO step.

In the other proposed notation systems aside from native fifths, a sharp corresponds to +1 EDO step, while a flat corresponds to -1. The Zarlino notation here uses the [[ternary]] Zarlino scale (see [[#Zarlino diatonic]]), or Ptolemy's intense diatonic, as its basic scale (which prioritizes the 5-limit, whereas native fifths prioritize 2.3.7), and uses to its advantage the fact that one step of 22edo maps to both 25/24 and 81/80 (a property of [[Porcupine]] temperament). Pajara uses the 10-note Pajara scale (see [[#Pajara]]) as its basis, which is generated by a perfect fifth but splits the octave in two to reach intervals of the full 7-limit relatively easily; the scale has four 2-step and one 3-step intervals per half-octave, which differ in size by a diatonic minor second, which is 1 step in 22edo.

The ADIN system uses the labels "nearminor" and "nearmajor" for intervals that 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. Note also that as "subminor" and "supermajor" intervals are the minor and major intervals of the diatonic MOS in 22edo, unqualified "major" and "minor" by default refer to these.

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 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
|55
|25/24, 28/27, [33/32], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|164
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218
|8/7, '''9/8''', 17/15
|D
|D#
|2
|(Super)major second
|-
|5
|273
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327
|6/5, 11/9
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|382
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|491
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545
|'''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
|655
|16/11
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|764
|[14/9], 11/7, 25/16
|Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|873
|5/3, 18/11
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|982
|'''7/4''', 16/9, 30/17
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1091
|'''[15/8]''', 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|10
|Octave
|}

== Structural theory ==
=== Temperaments and generators ===
==== Whole-octave temperaments ====
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. The temperament associated with this equivalence is '''Porcupine''', where 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.

==== Split-octave temperaments ====
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. 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]].

==== Temperaments of 11edo ====
Important temperaments that 22edo borrows from 11edo include [[Orgone]] (a 2.7.11 structure generated by the nearminor third, so that two of them form 16/11 and three form 7/4; note that this is in fact every other step of Porcupine), as well as [[Sentry]] (where two 9/7s reach 5/3, and in this case serves as every other step of Magic).

=== Tertian structure ===
22edo has four clear qualities of "thirds" that can serve as mediants in a chord bounded by a fifth. These are the subminor (273{{c}}, 5\22), nearminor (327{{c}}, 6\22), nearmajor (382{{c}}, 7\22), and supermajor (436{{c}}, 8\22) thirds, which reflect the intervals 7/6, 6/5, 5/4, and 9/7 respectively. As the gap between 6/5 and 5/4 is the same as that between 7/6 and 6/5 (or 5/4 and 9/7), 22edo's tertian structure is [[875/864|keemic]].

{| 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. 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.

=== Pythagorean diatonic ===
This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of 22edo. This is good for septimal harmony, not so much for 5-limit harmony (to get 5-limit thirds, you have to go 9 steps up!). A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone (in fact, it is enharmonic to the small whole-tone used in the Zarlino system). This can make interval classifications somewhat unintuitive if ups and downs are not used - for instance, a Nearmajor triad, 0-7-13, is C-D#-G (meaning that a third-sized interval is technically a second) - and in fact 22edo is the first edo other than 15edo (which is strange in its own right) to require ups and downs to notate all intervals without problems with intuition like this (Edos like 17 and 10 utilize semisharps and semiflats, which 22edo cannot use as it does not have neutral intervals.)

=== Zarlino diatonic ===
A faithful way of representing the 5-limit diatonic structure in 22edo (which, unlike its just counterpart, keeps the 7-limit convenient to access) is to use 5/4 as the scale's major third, and likewise 5/3 and 15/8 as the major sixth and seventh respectively. The scale pattern for zarlino is 4-3-2-4-3-4-2. 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), however it does have one thing going for it: in 22edo in particular (and a family of edos including 15, 29, and 37), the interval which separates Pythagorean intervals from their 5-limit counterparts is the exact same as the interval separating 5-limit major and minor intervals. This means that, if # and b are used to represent this interval as an accidental, no additional accidentals are necessary. This is why the 3-step interval is a "Nearmajor second" along with being a chromatic semitone.

=== Pajara ===
Note that the 1-step interval which serves as the 3-limit diatonic semitone is the very same as the 5-limit chromatic semitone, and that the 3-step interval serving as the 3-limit chromatic semitone also happens to map to a certain kind of large 5-limit diatonic semitone (such that it and the chromatic semitone stack to the 4-step whole tone). This suggests that if there was a way to swap the diatonic and chromatic semitones, we could faithfully represent the full 7-limit, including 5.

And as it turns out, there is a way to do that. We start with the pentatonic scale, not only because it's closer to even but because the interval between its small and large steps is exactly the 1-step interval we want to function as our chroma. Then, we add another pentatonic scale that is offset by a tritone from the first one. This results in the decatonic "Pajara[10]" scale (3-2-2-2-2-3-2-2-2-2), where every note has a corresponding note a tritone apart. It solves the problem of representing intervals of both 5 and 7 by introducing three new ordinal classes to provide space for 7-limit intervals to fit on their own degrees of the scale. That way, 7/4 isn't a subminor seventh, it's a major version of the Pajara 8-step. One can even define a notation system for Pajara, wherein the notes are numbered 0-9 and # and b represent alterations by a single step. Pajara retains the property where most notes have a fifth over them.

To extend to the 11-limit, Pajara[12] (1-2-2-2-2-2-1-2-2-2-2-2) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step.

==== Symmetric scale ====
One possible scale of 22edo, as mentioned previously, is the Pajara[10] decatonic scale, represented as {{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}. This scale can be explored [https://sevish.com/scaleworkshop/?n=22EDO%20jaric%20ssLssssLss%204%7C4%282%29%20soft&l=2Bm_4Bm_7Bm_9Bm_bBm_dBm_fBm_iBm_kBm_mBm&c=&w=r&a=i&y=pi&s=0&r=2o&b=hs&g=gh&version=2.5.7 here]. Below is a chart of its five modes, ordered by rotation. Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
!Mode on fifth
!Mode on fourth
|-
|Dynamic minor
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|
|
|-
|Static minor
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|major
|
|
|-
|Static major
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|
|
|-
|Dynamic major
|{{Interval ruler|22|0, 100, 250, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|
|
|-
|Augmented
|{{Interval ruler|22|0, 150, 250, 400, 500, 600, 750, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
|
|
|}

==== Pentachordal scale ====
This scale is constructed from two identical "pentachords" and the semioctave, and is represented as {{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}. Unlike the symmetric scale, there are ten distinct modes of the pentachordal scale. Ordered by rotation, they are as follows:
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
!Mode on fifth
!Mode on fourth
|-
|Bediyic
|{{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|Hininic
| -
|-
|Alternate minor (Skoronic)
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 950, 1100, 1200}}
|minor
|minor
|dim
|perfect
|minor
|major
|Aujalic
| -
|-
|Moriolic
|{{Interval ruler|22|0, 100, 200, 320, 500, 600, 700, 830, 990, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|major
|major
|Mielauic
|Hininic
|-
|Standard major (Staimosic)
|{{Interval ruler|22|0, 100, 200, 370, 500, 600, 700, 870, 990, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|Prathuic
|Aujalic
|-
|Sebaic
|{{Interval ruler|22|0, 100, 270, 370, 500, 600, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
| -
|Mielauic
|-
|Awanic
|{{Interval ruler|22|0, 170, 270, 370, 500, 670, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
| -
|Prathuic
|-
|Standard minor (Hininic)
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|minor
|Moriolic
|Bediyic
|-
|Aujalic
|{{Interval ruler|22|0, 100, 200, 360, 500, 600, 700, 800, 920, 1110, 1200}}
|minor
|major
|perfect
|perfect
|minor
|major
|Staimosic
|Skoronic
|-
|Alternate major (Kielauic)
|{{Interval ruler|22|0, 100, 270, 360, 500, 600, 700, 800, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|Sebaic
|Moriolic
|-
|Prathuic
|{{Interval ruler|22|0, 170, 270, 360, 500, 600, 700, 870, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|Awanic
|Staimosic
|}
Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].

=== Blackdye ===
Blackdye is a rank-3 scale similar to zarlino diatonic, which attempts to compromise between zarlino and Pythagorean diatonic in a way, by including a few intervals from both at once. Blackdye can be thought of as dividing each 4-step whole tone into a 3-step tone and a single step called an [[aberrisma]], which separates zarlino and Pythagorean intervals. The blackdye scale pattern is 3-1-3-2-3-1-3-1-3-2. One way to use blackdye is to essentially treat it as multiple overlapping diatonics, which one can modulate between.

=== Other scales ===
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 170, 330, 500, 700, 870, 1030, 1200}}
|Approximate Greek scale (equable diatonic), basic MOS of Porcupine.
|-
|Gramitonic (4L5s)
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|Basic MOS of Orwell temperament.
|-
|Zarlino diatonic
|{{Interval ruler|22|0, 110, 330, 500, 700, 800, 1030, 1200}}
|Greek scale (intense diatonic). Zarlino rank-3 diatonic.
|-
|Mosdiatonic
|{{Interval ruler|22|0, 50, 270, 500, 700, 750, 970, 1200}}
|Greek scale (Pythagorean or Archytas diatonic). Basic MOS of Superpyth.
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Pentic
|{{Interval ruler|22|0, 270, 500, 700, 970, 1200}}
|Basic MOS of Superpyth
|}

== Triads and tetrads ==

=== Triads bounded by P5 ===
{| class="wikitable"
!Name
!1
!2
!Bounding interval
!Edostep
!Chart
|-
|Sus4 triad
|Perfect 4th
|Supermajor 2nd
|Perfect 5th
|[0 9 13]
|{{Interval ruler|22|0, 500, 700}}
|-
|Supermajor triad
|Supermajor 3rd
|Subminor 3rd
|Perfect 5th
|[0 8 13]
|{{Interval ruler|22|0, 430, 700}}
|-
|Nearmajor triad
|Nearmajor 3rd
|Nearminor 3rd
|Perfect 5th
|[0 7 13]
|{{Interval ruler|22|0, 380, 700}}
|-
|Nearminor triad
|Nearminor 3rd
|Nearmajor 3rd
|Perfect 5th
|[0 6 13]
|{{Interval ruler|22|0, 320, 700}}
|-
|Subminor triad
|Subminor 3rd
|Supermajor 3rd
|Perfect 5th
|[0 5 13]
|{{Interval ruler|22|0, 270, 700}}
|-
|Sus2 triad
|Supermajor 2nd
|Perfect 4th
|Perfect 5th
|[0 4 13]
|{{Interval ruler|22|0, 200, 700}}
|}

=== Tetrads with P5th ===

==== Harmonic tetrads ====
These represent the conventional structure of a 4:5:6:7 harmonic tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect median above the middle note. This is the characteristic kind of chord found in Pajara temperament, and can also be thought of the stacking of a fifth-bounded triad and a fourth-bounded triad.
{| class="wikitable"
!Name
!1
!2
!3
!4
!Bounding interval 1
!Bounding interval 2
!Bounding interval 3
!Edostep
!Chart
|-
|Nearmajor harmonic tetrad
|Nearmajor 3rd
|Nearminor 3rd
|Subminor 3rd
|Supermajor 2nd
|Perfect 5th
|Subminor 7th
|Perfect 8ve
|[0 7 13 18]
|{{Interval ruler|22|0, 380, 700, 980, 1200}}
|-
|Nearminor harmonic tetrad
|Nearminor 3rd
|Nearmajor 3rd
|Supermajor 2nd
|Subminor 3rd
|Perfect 5th
|Supermajor 6th
|Perfect 8ve
|[0 6 13 17]
|{{Interval ruler|22|0, 320, 700, 920, 1200}}
|}

==== Diatonic tetrads ====
These represent the conventional structure of an 8:10:12:15 diatonic major tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect fifth above the middle note. The sus4 diatonic tetrad is the same as the triad version up to inversion.
{| class="wikitable"
!Name
!1
!2
!3
!Bounding interval 1
!Bounding interval 2
!Edostep
!Chart
|-
|Supermajor diatonic tetrad
|Supermajor 3rd
|Subminor 3rd
|Supermajor 3rd
|Perfect 5th
|Supermajor 7th
|[0 8 13 21]
|{{Interval ruler|22|0, 430, 700, 1150, 1200}}
|-
|Nearmajor diatonic tetrad
|Nearmajor 3rd
|Nearminor 3rd
|Nearmajor 3rd
|Perfect 5th
|Nearmajor 7th
|[0 7 13 20]
|{{Interval ruler|22|0, 380, 700, 1080, 1200}}
|-
|Nearminor diatonic tetrad
|Nearminor 3rd
|Nearmajor 3rd
|Nearminor 3rd
|Perfect 5th
|Nearminor 7th
|[0 6 13 19]
|{{Interval ruler|22|0, 320, 700, 1020, 1200}}
|-
|Subminor diatonic tetrad
|Subminor 3rd
|Supermajor 3rd
|Subminor 3rd
|Perfect 5th
|Subminor 7th
|[0 5 13 18]
|{{Interval ruler|22|0, 270, 700, 970, 1200}}
|-
|Sus2 diatonic tetrad
|Supermajor 2nd
|Perfect 4th
|Supermajor 2nd
|Perfect 5th
|Supermajor 6th
|[0 4 13 17]
|{{Interval ruler|22|0, 210, 700, 920, 1200}}
|}

{{Cat|Edos}}

22edo

2026-02-12T06:23:15Z

Hkm: shifting info from edostep interpretations/tempered commas list to intervals list. is this more helpful? i would think so. perhaps people disagree, and due to that possibility i have separated these changes off from any future ones so that this can easily be reverted

'''22edo''', or 22 equal divisions of the octave, 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 [[9/8]] and [[8/7]] are the same interval. 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.

22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]], which ADIN calls "nearminor" and "nearmajor" thirds. As a result, 22 is perhaps the smallest EDO that can be considered to include 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.

== General theory ==
=== 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)

=== 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 of an octave.
* [[55/54]] (telepath), equating 6/5 with 11/9
* [[64/63]] (archytas), equating 16/9 with 7/4
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7
* [[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)

[[Regular temperament]]s associated with these are discussed in [[#Temperaments and generators]]. 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.}}

=== JI approximation ===
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the 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.

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.

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 and notation ===
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).

The "native-fifths" system mentioned here is the most commonly used system, and the one that most microtonal notation systems support by default. It is derived through stacking 22edo's tempered version of 3/2 and assigning names accordingly. As a result, the nominals (C, D, E, F, G, ...) follow the diatonic MOS, where the distances between the large steps (C-D, D-E, F-G, G-A, A-B) are 4 EDO steps, and the distances between the small steps (E-F, B-C) are 1 EDO step. Therefore, a sharp corresponds to +3 EDO steps while a flat corresponds to -3 (representing the diatonic chroma in each case). Each sharp or flat can be split into three distinct notes, so we use the accidental ^ to raise by one EDO step and v to lower by one EDO step.

In the other proposed notation systems aside from native fifths, a sharp corresponds to +1 EDO step, while a flat corresponds to -1. The Zarlino notation here uses the [[ternary]] Zarlino scale (see [[#Zarlino diatonic]]), or Ptolemy's intense diatonic, as its basic scale (which prioritizes the 5-limit, whereas native fifths prioritize 2.3.7), and uses to its advantage the fact that one step of 22edo maps to both 25/24 and 81/80 (a property of [[Porcupine]] temperament). Pajara uses the 10-note Pajara scale (see [[#Pajara]]) as its basis, which is generated by a perfect fifth but splits the octave in two to reach intervals of the full 7-limit relatively easily; the scale has four 2-step and one 3-step intervals per half-octave, which differ in size by a diatonic minor second, which is 1 step in 22edo.

The ADIN system uses the labels "nearminor" and "nearmajor" for intervals that 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. Note also that as "subminor" and "supermajor" intervals are the minor and major intervals of the diatonic MOS in 22edo, unqualified "major" and "minor" by default refer to these.

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 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
|55
|25/24, 28/27, [33/32], 36/35
|^C, Db
|C#
|1b
|(Sub)minor second
|-
|2
|109
|[16/15], 15/14, 18/17, ['''17/16''']
|vC#, ^Db
|Db
|1
|Nearminor second
|-
|3
|164
|10/9, [11/10], 12/11
|C#, vD
|D
|1#
|Nearmajor second
|-
|4
|218
|8/7, '''9/8''', 17/15
|D
|D#
|2
|(Super)major second
|-
|5
|273
|7/6
|^D, Eb
|Ebb / Dx
|2#
|(Sub)minor third
|-
|6
|327
|6/5, 11/9
|vD#, ^Eb
|Eb
|3b
|Nearminor third
|-
|7
|382
|'''[5/4]'''
|D#, vE
|E
|3
|Nearmajor third
|-
|8
|436
|[9/7], 14/11, 32/25
|E
|E#
|4b
|(Super)major third
|-
|9
|491
|4/3
|F
|F
|4
|Perfect fourth
|-
|10
|545
|'''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
|655
|16/11
|F#, vG
|Gb
|6b
|Near fifth
|-
|13
|709
|'''3/2'''
|G
|G
|6
|Perfect fifth
|-
|14
|764
|[14/9], 11/7, 25/16
|Ab
|G#
|6#
|(Sub)minor sixth
|-
|15
|818
|[8/5]
|vG#, ^Ab
|Ab
|7
|Nearminor sixth
|-
|16
|873
|5/3, 18/11
|G#, vA
|A
|7#
|Nearmajor sixth
|-
|17
|927
|12/7
|A
|A#
|8b
|(Super)major sixth
|-
|18
|982
|'''7/4''', 16/9, 30/17
|^A, Bb
|Bbb / Ax
|8
|(Sub)minor seventh
|-
|19
|1036
|9/5, [20/11], 11/6
|vA#, ^Bb
|Bb
|9b
|Nearminor seventh
|-
|20
|1091
|'''[15/8]''', 28/15, 17/9, [32/17]
|A#, vB
|B
|9
|Nearmajor seventh
|-
|21
|1145
|48/25, 27/14, [64/33], 35/18
|B
|Cb
|9#
|(Super)major seventh
|-
|22
|1200
|2/1
|C
|C
|10
|Octave
|}

== Structural theory ==
=== Temperaments and generators ===
==== Whole-octave temperaments ====
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. The temperament associated with this equivalence is '''Porcupine''', where 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.

The temperament generated by 1\22, known as '''[[Escapade]]''', therefore splits this structure further into three. Three generators represent 11/10 (but not 10/9 or 12/11, outside of 22edo itself); seven represent 5/4 and nine represent 4/3. Therefore, the generator itself represents sqrt(16/15), identifiable as [[33/32]], 32/31, and [[31/30]]. This structure is shared with EDOs like [[43edo|43]], [[65edo|65]], and [[87edo]].

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.

==== Split-octave temperaments ====
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. 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]].

==== Temperaments of 11edo ====
Important temperaments that 22edo borrows from 11edo include [[Orgone]] (a 2.7.11 structure generated by the nearminor third, so that two of them form 16/11 and three form 7/4; note that this is in fact every other step of Porcupine), as well as [[Sentry]] (where two 9/7s reach 5/3, and in this case serves as every other step of Magic).

=== Tertian structure ===
22edo has four clear qualities of "thirds" that can serve as mediants in a chord bounded by a fifth. These are the subminor (273{{c}}, 5\22), nearminor (327{{c}}, 6\22), nearmajor (382{{c}}, 7\22), and supermajor (436{{c}}, 8\22) thirds, which reflect the intervals 7/6, 6/5, 5/4, and 9/7 respectively. As the gap between 6/5 and 5/4 is the same as that between 7/6 and 6/5 (or 5/4 and 9/7), 22edo's tertian structure is [[875/864|keemic]].

{| 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. 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.

=== Pythagorean diatonic ===
This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of 22edo. This is good for septimal harmony, not so much for 5-limit harmony (to get 5-limit thirds, you have to go 9 steps up!). A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone (in fact, it is enharmonic to the small whole-tone used in the Zarlino system). This can make interval classifications somewhat unintuitive if ups and downs are not used - for instance, a Nearmajor triad, 0-7-13, is C-D#-G (meaning that a third-sized interval is technically a second) - and in fact 22edo is the first edo other than 15edo (which is strange in its own right) to require ups and downs to notate all intervals without problems with intuition like this (Edos like 17 and 10 utilize semisharps and semiflats, which 22edo cannot use as it does not have neutral intervals.)

=== Zarlino diatonic ===
A faithful way of representing the 5-limit diatonic structure in 22edo (which, unlike its just counterpart, keeps the 7-limit convenient to access) is to use 5/4 as the scale's major third, and likewise 5/3 and 15/8 as the major sixth and seventh respectively. The scale pattern for zarlino is 4-3-2-4-3-4-2. 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), however it does have one thing going for it: in 22edo in particular (and a family of edos including 15, 29, and 37), the interval which separates Pythagorean intervals from their 5-limit counterparts is the exact same as the interval separating 5-limit major and minor intervals. This means that, if # and b are used to represent this interval as an accidental, no additional accidentals are necessary. This is why the 3-step interval is a "Nearmajor second" along with being a chromatic semitone.

=== Pajara ===
Note that the 1-step interval which serves as the 3-limit diatonic semitone is the very same as the 5-limit chromatic semitone, and that the 3-step interval serving as the 3-limit chromatic semitone also happens to map to a certain kind of large 5-limit diatonic semitone (such that it and the chromatic semitone stack to the 4-step whole tone). This suggests that if there was a way to swap the diatonic and chromatic semitones, we could faithfully represent the full 7-limit, including 5.

And as it turns out, there is a way to do that. We start with the pentatonic scale, not only because it's closer to even but because the interval between its small and large steps is exactly the 1-step interval we want to function as our chroma. Then, we add another pentatonic scale that is offset by a tritone from the first one. This results in the decatonic "Pajara[10]" scale (3-2-2-2-2-3-2-2-2-2), where every note has a corresponding note a tritone apart. It solves the problem of representing intervals of both 5 and 7 by introducing three new ordinal classes to provide space for 7-limit intervals to fit on their own degrees of the scale. That way, 7/4 isn't a subminor seventh, it's a major version of the Pajara 8-step. One can even define a notation system for Pajara, wherein the notes are numbered 0-9 and # and b represent alterations by a single step. Pajara retains the property where most notes have a fifth over them.

To extend to the 11-limit, Pajara[12] (1-2-2-2-2-2-1-2-2-2-2-2) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step.

==== Symmetric scale ====
One possible scale of 22edo, as mentioned previously, is the Pajara[10] decatonic scale, represented as {{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}. This scale can be explored [https://sevish.com/scaleworkshop/?n=22EDO%20jaric%20ssLssssLss%204%7C4%282%29%20soft&l=2Bm_4Bm_7Bm_9Bm_bBm_dBm_fBm_iBm_kBm_mBm&c=&w=r&a=i&y=pi&s=0&r=2o&b=hs&g=gh&version=2.5.7 here]. Below is a chart of its five modes, ordered by rotation. Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
!Mode on fifth
!Mode on fourth
|-
|Dynamic minor
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|
|
|-
|Static minor
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|major
|
|
|-
|Static major
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|
|
|-
|Dynamic major
|{{Interval ruler|22|0, 100, 250, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|
|
|-
|Augmented
|{{Interval ruler|22|0, 150, 250, 400, 500, 600, 750, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
|
|
|}

==== Pentachordal scale ====
This scale is constructed from two identical "pentachords" and the semioctave, and is represented as {{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}. Unlike the symmetric scale, there are ten distinct modes of the pentachordal scale. Ordered by rotation, they are as follows:
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
!Mode on fifth
!Mode on fourth
|-
|Bediyic
|{{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|Hininic
| -
|-
|Alternate minor (Skoronic)
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 950, 1100, 1200}}
|minor
|minor
|dim
|perfect
|minor
|major
|Aujalic
| -
|-
|Moriolic
|{{Interval ruler|22|0, 100, 200, 320, 500, 600, 700, 830, 990, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|major
|major
|Mielauic
|Hininic
|-
|Standard major (Staimosic)
|{{Interval ruler|22|0, 100, 200, 370, 500, 600, 700, 870, 990, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|Prathuic
|Aujalic
|-
|Sebaic
|{{Interval ruler|22|0, 100, 270, 370, 500, 600, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
| -
|Mielauic
|-
|Awanic
|{{Interval ruler|22|0, 170, 270, 370, 500, 670, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
| -
|Prathuic
|-
|Standard minor (Hininic)
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|minor
|Moriolic
|Bediyic
|-
|Aujalic
|{{Interval ruler|22|0, 100, 200, 360, 500, 600, 700, 800, 920, 1110, 1200}}
|minor
|major
|perfect
|perfect
|minor
|major
|Staimosic
|Skoronic
|-
|Alternate major (Kielauic)
|{{Interval ruler|22|0, 100, 270, 360, 500, 600, 700, 800, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|Sebaic
|Moriolic
|-
|Prathuic
|{{Interval ruler|22|0, 170, 270, 360, 500, 600, 700, 870, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|Awanic
|Staimosic
|}
Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].

=== Blackdye ===
Blackdye is a rank-3 scale similar to zarlino diatonic, which attempts to compromise between zarlino and Pythagorean diatonic in a way, by including a few intervals from both at once. Blackdye can be thought of as dividing each 4-step whole tone into a 3-step tone and a single step called an [[aberrisma]], which separates zarlino and Pythagorean intervals. The blackdye scale pattern is 3-1-3-2-3-1-3-1-3-2. One way to use blackdye is to essentially treat it as multiple overlapping diatonics, which one can modulate between.

=== Other scales ===
{| class="wikitable"
!Name
!Chart
!Notes
|-
|Onyx
|{{Interval ruler|22|0, 170, 330, 500, 700, 870, 1030, 1200}}
|Approximate Greek scale (equable diatonic), basic MOS of Porcupine.
|-
|Gramitonic (4L5s)
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}
|Basic MOS of Orwell temperament.
|-
|Zarlino diatonic
|{{Interval ruler|22|0, 110, 330, 500, 700, 800, 1030, 1200}}
|Greek scale (intense diatonic). Zarlino rank-3 diatonic.
|-
|Mosdiatonic
|{{Interval ruler|22|0, 50, 270, 500, 700, 750, 970, 1200}}
|Greek scale (Pythagorean or Archytas diatonic). Basic MOS of Superpyth.
|-
|Zarlino pentatonic
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}
|One possible pentatonic analog to the Zarlino diatonic.
|-
|Pentic
|{{Interval ruler|22|0, 270, 500, 700, 970, 1200}}
|Basic MOS of Superpyth
|}

== Triads and tetrads ==

=== Triads bounded by P5 ===
{| class="wikitable"
!Name
!1
!2
!Bounding interval
!Edostep
!Chart
|-
|Sus4 triad
|Perfect 4th
|Supermajor 2nd
|Perfect 5th
|[0 9 13]
|{{Interval ruler|22|0, 500, 700}}
|-
|Supermajor triad
|Supermajor 3rd
|Subminor 3rd
|Perfect 5th
|[0 8 13]
|{{Interval ruler|22|0, 430, 700}}
|-
|Nearmajor triad
|Nearmajor 3rd
|Nearminor 3rd
|Perfect 5th
|[0 7 13]
|{{Interval ruler|22|0, 380, 700}}
|-
|Nearminor triad
|Nearminor 3rd
|Nearmajor 3rd
|Perfect 5th
|[0 6 13]
|{{Interval ruler|22|0, 320, 700}}
|-
|Subminor triad
|Subminor 3rd
|Supermajor 3rd
|Perfect 5th
|[0 5 13]
|{{Interval ruler|22|0, 270, 700}}
|-
|Sus2 triad
|Supermajor 2nd
|Perfect 4th
|Perfect 5th
|[0 4 13]
|{{Interval ruler|22|0, 200, 700}}
|}

=== Tetrads with P5th ===

==== Harmonic tetrads ====
These represent the conventional structure of a 4:5:6:7 harmonic tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect median above the middle note. This is the characteristic kind of chord found in Pajara temperament, and can also be thought of the stacking of a fifth-bounded triad and a fourth-bounded triad.
{| class="wikitable"
!Name
!1
!2
!3
!4
!Bounding interval 1
!Bounding interval 2
!Bounding interval 3
!Edostep
!Chart
|-
|Nearmajor harmonic tetrad
|Nearmajor 3rd
|Nearminor 3rd
|Subminor 3rd
|Supermajor 2nd
|Perfect 5th
|Subminor 7th
|Perfect 8ve
|[0 7 13 18]
|{{Interval ruler|22|0, 380, 700, 980, 1200}}
|-
|Nearminor harmonic tetrad
|Nearminor 3rd
|Nearmajor 3rd
|Supermajor 2nd
|Subminor 3rd
|Perfect 5th
|Supermajor 6th
|Perfect 8ve
|[0 6 13 17]
|{{Interval ruler|22|0, 320, 700, 920, 1200}}
|}

==== Diatonic tetrads ====
These represent the conventional structure of an 8:10:12:15 diatonic major tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect fifth above the middle note. The sus4 diatonic tetrad is the same as the triad version up to inversion.
{| class="wikitable"
!Name
!1
!2
!3
!Bounding interval 1
!Bounding interval 2
!Edostep
!Chart
|-
|Supermajor diatonic tetrad
|Supermajor 3rd
|Subminor 3rd
|Supermajor 3rd
|Perfect 5th
|Supermajor 7th
|[0 8 13 21]
|{{Interval ruler|22|0, 430, 700, 1150, 1200}}
|-
|Nearmajor diatonic tetrad
|Nearmajor 3rd
|Nearminor 3rd
|Nearmajor 3rd
|Perfect 5th
|Nearmajor 7th
|[0 7 13 20]
|{{Interval ruler|22|0, 380, 700, 1080, 1200}}
|-
|Nearminor diatonic tetrad
|Nearminor 3rd
|Nearmajor 3rd
|Nearminor 3rd
|Perfect 5th
|Nearminor 7th
|[0 6 13 19]
|{{Interval ruler|22|0, 320, 700, 1020, 1200}}
|-
|Subminor diatonic tetrad
|Subminor 3rd
|Supermajor 3rd
|Subminor 3rd
|Perfect 5th
|Subminor 7th
|[0 5 13 18]
|{{Interval ruler|22|0, 270, 700, 970, 1200}}
|-
|Sus2 diatonic tetrad
|Supermajor 2nd
|Perfect 4th
|Supermajor 2nd
|Perfect 5th
|Supermajor 6th
|[0 4 13 17]
|{{Interval ruler|22|0, 210, 700, 920, 1200}}
|}

{{Cat|Edos}}

User:Hkm/SEO Goals

2026-02-02T04:11:41Z

Hkm:

- rename Homepage title to "Xenharmonic Reference" rather than "XenReference"

- Submit a sitemap.xml

- <s>get clean links</s>

- do we have "canonical" tags in our HTML? No we don't. That's why Google is indexing our main page twice. https://www.mediawiki.org/wiki/Manual:$wgEnableCanonicalServerLink

- <s>Remove the statement on the main page about the wiki being "mostly empty at the moment"</s>

- We need *some* backlinks. Maybe write about the site in a community post on YouTube, or post about it on Reddit?

- get "noindex" on user pages

- get more citations (to get that credibility up)

User:Hkm/SEO Goals

2026-01-30T00:27:26Z

Hkm:

- rename Homepage title to "Xenharmonic Reference" rather than "XenReference"

- Submit a sitemap.xml

- <s>get clean links</s>

- do we have "canonical" tags in our HTML?

- <s>Remove the statement on the main page about the wiki being "mostly empty at the moment"</s>

- We need *some* backlinks. Maybe write about the site in a community post on YouTube, or post about it on Reddit?

- get "noindex" on user pages

- get more citations (to get that credibility up)

Combination product set

2026-01-30T00:26:19Z

Hkm:

{{Problematic}}
{{Expert}}
A '''combination product set''' (CPS) is a scale (usually [[Just intonation|JI]]) generated by the following procedure:

# A list of chosen intervals (usually odd harmonics) is the starting point.
# All the combinations of some number of distinct intervals from the list are obtained. The same number of intervals is used for every combination.
# Each of the above combinations of intervals is stacked together into one interval.
# This results in a list of notes. One note is chosen as the tonic.
# The resulting intervals relative to the tonic are octave-reduced.

CPSes are notable as ''every'' note of a CPS has a different JI chord on it.

A CPS is a subset of an iterated [[cross-set]] of a chord with itself. Specifically, a CPS of chord X is the subset of all elements of cross-set(X, cross-set(X, cross-set(...))) that do not stack any ratio of X more than once. CPSes were developed by Erv Wilson.
== Example (1, 3, 5, 7 hexany) ==
# In this example we choose four odd harmonics: 1, 3, 5, 7.
# We get all combinations of 2 different odd harmonics: [1, 3], [1, 5], [1, 7], [3, 5], [3, 7], [5, 7].
# For each combination of intervals, stack the intervals together: 3, 5, 7, 15, 21, 35.
# Choose 3 as the tonic. (This choice just amounts to choosing a mode of the final scale.)
# Measure all the other notes relative to the chosen tonic: 1/1, 5/3, 7/3, 15/3 = 5/1, 21/3 = 7/1, 35/3.
# Octave-reduce everything: 1/1, 5/3, 7/6, 5/4, 7/4, 35/24.
This results in the 6-note scale [1/1, 7/6, 5/4, 35/24, 5/3, 7/4, 2/1], hence "hexany".

== Types of CPSes ==
Common sizes for CPSes have specific names:
# '''Hexany''': Choose 2 out of a list of 4 intervals
#* '''Stellated hexany''': A hexany combined with combinations of 1 and combinations of 3
#* '''Bihexany''': Superimposition of two offset hexanies
# '''Dekany''': Choose 2 (or 3) out of a list of 5 intervals
# '''Pentadekany''': Choose 2 (or 4) out of a list of 6 intervals
# '''Eikosany''': Choose 3 out of a list of 6 intervals, creating a 20-note scale

== Playing with CPSes ==
ScaleWorkshop 3 allows you to make JI CPSes:
# Go to https://scaleworkshop.plainsound.org
# Click "New scale"
# Select "Combination product set"

== Music ==
* [https://youtu.be/62FggGa9bMg?si=fC72fRe9f8Zk8OcA Eikosany Study - Daniel Corral (1 3 5 7 9 11 eikosany)]
* [https://youtu.be/AWeFIOwGBh8?si=w1ddUi3LjNLTAsHk Nodial - Outward Once (an eikosany)]
{{Cat|Scale construction}}

Glossary

2026-01-28T04:58:56Z

Hkm: Undo revision 3057 by Vector (talk). We had a talk about this on XR; that sense is proscribed for a good reason

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

== 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

== 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
== 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.

Categories: Core knowledge
== Extension ==
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 often 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. {{Adv|If you don't interpret the new intervals of a weak extension, the result is called ''contorsion''.}}

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 [[XenReference:Guidelines]].''

Categories: RTT, Somewhat technical

== 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 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'' 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.

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 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 more commonly means prime-limit.

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 Xen 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
== Natave ==
The frequency ratio ''e''/1. Sometimes called a ''neper''. Mainly used in theoretical xen math.

== 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

== 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).
** 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 the equal division that supports it to be divisible by some number (namely, the number of 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. This is what makes a regular temperament "regular".

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 XenReference, ''scales are periodic unless stated otherwise.'' 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

== 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]]. 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 [[Mathematics of commas#Square superparticulars|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
== Union ==
The '''union''' of two scales/chords is a scale/chord with all pitches that occur in either scale/chord. In other words, it's a shorter way of saying "superimposition". If you change the offset between the two scales/chords, taking their union usually yields a different scale/chord.

Examples:
* A [[cross-set]] is a union of copies of the same scale placed on different offsets.
** 15edo pentawood uses two copies of 5edo offset by 1\15; 20edo pentawood uses two copies of 5edo offset by 1\20.
* ''Polysystemic tuning'' uses a union of multiple systems, for example 5edo and 7edo.

Categories: Math terms

== 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.

'''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.

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) or strict variety (SV). For example, [[MOS]] scales can be defined as scales that are MV2.

Categories: Scales

41edo

2026-01-28T04:57:30Z

Hkm: /* JI approximation */

'''41edo''', or 41 equal divisions of the octave, is an equal tuning with a step size of approximately 29 cents. It is known for its relatively good approximation of 11-limit just intonation.

== Theory ==

==== JI approximation ====
41edo is most accurately a 2.3.5.7.11 tuning, though it also has an acceptable if sharp 13th harmonic, notably widening the difference between the [[Collection of chords#Arto triad|arto]] (10:13:15) and [[Collection of chords#Tendo triad|tendo]] (1/10:1/13:1/15) triads such that they become simple [[Slendric]] divisions of the fifth. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 7-6-4-7-6-7-4. However, it also features a MOS diatonic of 7-7-3-7-7-7-3.

{{Harmonics in ED|41|31|0}}
{| class="wikitable"
|+Thirds in 41edo
!Quality
|Subminor
|'''Novaminor'''
|Pentaminor
|Neutral
|Pentamajor
|'''Novamajor'''
|Supermajor
|-
!Cents
|263
|'''293'''
|322
|351
|381
|'''410'''
|439
|-
!Just interpretation
|7/6
|'''13/11'''
|6/5
|11/9
|5/4
|'''14/11'''
|9/7
|}
Thirds available in the diatonic scale generated by stacking the perfect fifth are bolded.

=== Chords ===
41edo has three different flavors of minor and major intervals as well as neutral intervals. Its subminor and supermajor intervals approximate simpler septimal ratios such as 7/4 and 9/7, while its supraminor ("pentaminor") and submajor ("pentamajor") intervals approximate classical 5-limit harmony which includes ratios like 5/4 and 9/5, and its plain major and plain minor intervals approximate classic 3-limit ratios. As a result, 41edo has nine qualities of tertian, fifth-bounded triad: tendo, supermajor, novamajor, pentamajor, neutral, pentaminor, novaminor, subminor. However, 41edo lacks true interseptimal intervals (to reach a tuning with both neutrals and interseptimals, 58edo must be used), so as for latal fourth-bounded triads, there are only four qualities.

=== Scales ===
41edo's 5-limit intervals are not found particularly early on in the chain of fifths, with 6/5 being an augmented second and 5/4 a diminished fourth. Notably, 41edo has a 17-note chromatic scale generated by the perfect fifth, 3-3-3-1-3-3-1-3-3-3-1-3-3-1-3-3-1, in which the classical (~5/4) major and classical (~6/5) minor thirds span the same number of scale steps, giving the scale familiar chord qualities.

=== Regular temperaments ===
41edo shares Schismic (and its extension Garibaldi, and thus Marvel and Hemifamity) with 29edo, Slendric (and its extension Miracle) with 31edo, Tetracot with 34edo, and Magic with 22edo. Magic is especially important here as it forms the fret layout and main string tuning for the Kite guitar.

It also contains a slightly-stretched version of equal Bohlen-Pierce tuning (where the perfect twelfth of 3/1 is split into 13 equal parts) via every fifth step. If used in a linear temperament as the generator, this temperament is called Bohpier.

== Notation ==
Since 41edo has a perfect fifth which is split exactly in half, semisharps and semiflats (as in [[Diatonic notation|neutral diatonic notation]]) can be used to notate it. A useful addition is ups and downs, which naturally reflect 41edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor. (In fact, "up" can be declared equivalent to "super"/"supra" and "down" equivalent to "sub".)

== Practice ==
41edo is used by the musician and conlanger Lamplight as a standard tuning for their ''Shasavic'' theory of music.

41edo is used in Kite Giedraitis's Kite Guitar, which manages the high density of notes by only having frets for every other note of 41edo, and tuning the strings in a way that allows one string to fill in the gaps of an adjacent string.

{{Cat|Edos}}

Archy

2026-01-27T23:20:21Z

Hkm: fix garden path

'''Archy''' is the temperament that tempers out the '''archytas comma,''' 64/63, equating [[2.3.7 subgroup|septal]] intervals with nearby [[Pythagorean tuning|diatonic]] ones. In Archy, the generator is a fourth, the period is an octave, and 2 flattened [[Perfect fourth|fourths]] of about 490 cents stack to a sharply tuned [[Septal subminor seventh|7th harmonic]] of about 980 cents. Equivalently, the pythagorean (9/8) major second is mapped to the same pitch as the septimal (8/7) major second.

Archy is usually tuned such that the subminor (7/6) third is close to accurately tuned; flatter tunings of the fourth lead to more accurate tunings of the 7th harmonic, at the cost of the usability of the diatonic scale. The tuning that justly tunes the harmonic seventh places the perfect fourth at 484.4 cents, which leads to a diatonic scale with a small step at the uncomfortable size of 22 cents.

As a monocot temperament (a temperament generated by a perfect fourth or fifth), Archy can be notated with standard [[Diatonic notation|diatonic]] notation. However, this is somewhat awkward, as Archy is more cleanly analyzed as a 5-form temperament, producing an [[equipentatonic]] scale, so perhaps diamond-MOS or KISS notation with [[pentic]] would be better suited for it.

== Structural theory ==

=== Extensions ===
The following are extensions to prime 5 (i. e. ways to map intervals involving prime 5 onto the existing structure of 2.3.7 archy)

==== 5/4 as limma-flat major third (22 & 27), often called "Superpyth" ====
The canonical extension, equates 5/4 with the diatonic augmented second, or an octave-reduced stack of 9 fifths, which can be seen in the 5-form as a major third flattened by a diatonic semitone representing the [[septimal quartertone]] (36/35, the interval between 5/4 and 9/7) and the [[Meantone|syntonic comma]]. It can be seen as the 22 & 27 temperament. The preferred tuning range for the fifth in this extension tends to be somewhat flatter than that of archy; tunings where both the supermajor (9/7) and subminor (7/6) thirds are somewhat accurate are preferred.

==== 5/4 as doubly limma-flat major third (5 & 37) ====
This is an alternative extension, best tuned sharp of 27edo. Instead of flattening the major third by a diatonic semitone to reach the 5th harmonic, you flatten by two diatonic semitones. In diatonic notation, this means that 5/4 is the double-augmented unison.

== Compositional theory ==

=== Chords ===
In Archy, the diatonic major and minor chords essentially have their roles swapped from in meantone, as they now represent the [[Collection of chords|supermajor triad]] and [[Collection of chords|subminor triad]] respectively, and the minor chord is the more stable of the two. This can be seen by how the supermajor third is, in the 5-form, a flat fourth, serving a somewhat similar role to the diminished fifth in diatonic. The triad [0 4/3 7/4~14/9] is an important [[Collection of chords#Essentially tempered chords|essentially tempered chord]], although HKM finds that its other closed-voice inversions do not sound as if they contain septimal intervals unless the fifth is tuned as sharp as that of 37edo.

Due to existing in 2.3.7, Archy also supports the latal triads (bounded by a fourth, made from intervals near 250c, like 6:7:8), with 1/1-8/7-4/3 in particular appearing as part of the suspended tetrad.

{{cat|Temperaments}}

Archy

2026-01-27T22:28:04Z

Hkm: /* Chords */

'''Archy''' is the temperament that tempers out the '''archytas comma,''' 64/63, equating [[2.3.7 subgroup|septal]] intervals with nearby [[Pythagorean tuning|diatonic]] ones. In Archy, the generator is a fourth, the period is an octave, and 2 flattened [[Perfect fourth|fourths]] of about 490 cents stack to a sharply tuned [[Septal subminor seventh|7th harmonic]] of about 980 cents. Equivalently, the pythagorean (9/8) major second is mapped to the same pitch as the septimal (8/7) major second.

Archy is usually tuned such that the subminor (7/6) third is close to accurately tuned; flatter tunings of the fourth lead to more accurate tunings of the 7th harmonic, at the cost of the usability of the diatonic scale. The tuning that leaves the harmonic seventh just places the perfect fourth at 484.4 cents, which leads to a diatonic scale with a small step at the uncomfortable size of 22 cents.

As a monocot temperament (a temperament generated by a perfect fourth or fifth), Archy can be notated with standard [[Diatonic notation|diatonic]] notation. However, this is somewhat awkward, as Archy is more cleanly analyzed as a 5-form temperament, producing an [[equipentatonic]] scale, so perhaps diamond-MOS or KISS notation with [[pentic]] would be better suited for it.

== Structural theory ==

=== Extensions ===
The following are extensions to prime 5 (i. e. ways to map intervals involving prime 5 onto the existing structure of 2.3.7 archy)

==== 5/4 as limma-flat major third (22 & 27), often called "Superpyth" ====
The canonical extension, equates 5/4 with the diatonic augmented second, or an octave-reduced stack of 9 fifths, which can be seen in the 5-form as a major third flattened by a diatonic semitone representing the [[septimal quartertone]] (36/35, the interval between 5/4 and 9/7) and the [[Meantone|syntonic comma]]. It can be seen as the 22 & 27 temperament. The preferred tuning range for the fifth in this extension tends to be somewhat flatter than that of archy; tunings where both the supermajor (9/7) and subminor (7/6) thirds are somewhat accurate are preferred.

==== 5/4 as doubly limma-flat major third (5 & 37) ====
This is an alternative extension, best tuned sharp of 27edo. Instead of flattening the major third by a diatonic semitone to reach the 5th harmonic, you flatten by two diatonic semitones. In diatonic notation, this means that 5/4 is the double-augmented unison.

== Compositional theory ==

=== Chords ===
In Archy, the diatonic major and minor chords essentially have their roles swapped from in meantone, as they now represent the [[Collection of chords|supermajor triad]] and [[Collection of chords|subminor triad]] respectively, and the minor chord is the more stable of the two. This can be seen by how the supermajor third is, in the 5-form, a flat fourth, serving a somewhat similar role to the diminished fifth in diatonic. The triad [0 4/3 7/4~14/9] is an important [[Collection of chords#Essentially tempered chords|essentially tempered chord]], although HKM finds that its other closed-voice inversions do not sound as if they contain septimal intervals unless the fifth is tuned as sharp as that of 37edo.

Due to existing in 2.3.7, Archy also supports the latal triads (bounded by a fourth, made from intervals near 250c, like 6:7:8), with 1/1-8/7-4/3 in particular appearing as part of the suspended tetrad.

{{cat|Temperaments}}

Archy

2026-01-27T22:09:55Z

Hkm:

'''Archy''' is the temperament that tempers out the '''archytas comma,''' 64/63, equating [[2.3.7 subgroup|septal]] intervals with nearby [[Pythagorean tuning|diatonic]] ones. In Archy, the generator is a fourth, the period is an octave, and 2 flattened [[Perfect fourth|fourths]] of about 490 cents stack to a sharply tuned [[Septal subminor seventh|7th harmonic]] of about 980 cents. Equivalently, the pythagorean (9/8) major second is mapped to the same pitch as the septimal (8/7) major second.

Archy is usually tuned such that the subminor (7/6) third is close to accurately tuned; flatter tunings of the fourth lead to more accurate tunings of the 7th harmonic, at the cost of the usability of the diatonic scale. The tuning that leaves the harmonic seventh just places the perfect fourth at 484.4 cents, which leads to a diatonic scale with a small step at the uncomfortable size of 22 cents.

As a monocot temperament (a temperament generated by a perfect fourth or fifth), Archy can be notated with standard [[Diatonic notation|diatonic]] notation. However, this is somewhat awkward, as Archy is more cleanly analyzed as a 5-form temperament, producing an [[equipentatonic]] scale, so perhaps diamond-MOS or KISS notation with [[pentic]] would be better suited for it.

== Structural theory ==

=== Extensions ===
The following are extensions to prime 5 (i. e. ways to map intervals involving prime 5 onto the existing structure of 2.3.7 archy)

==== 5/4 as limma-flat major third (22 & 27), often called "Superpyth" ====
The canonical extension, equates 5/4 with the diatonic augmented second, or an octave-reduced stack of 9 fifths, which can be seen in the 5-form as a major third flattened by a diatonic semitone representing the [[septimal quartertone]] (36/35, the interval between 5/4 and 9/7) and the [[Meantone|syntonic comma]]. It can be seen as the 22 & 27 temperament. The preferred tuning range for the fifth in this extension tends to be somewhat flatter than that of archy; tunings where both the supermajor (9/7) and subminor (7/6) thirds are somewhat accurate are preferred.

==== 5/4 as doubly limma-flat major third (5 & 37) ====
This is an alternative extension, best tuned sharp of 27edo. Instead of flattening the major third by a diatonic semitone to reach the 5th harmonic, you flatten by two diatonic semitones. In diatonic notation, this means that 5/4 is the double-augmented unison.

== Compositional theory ==

=== Chords ===
In Archy, the diatonic major and minor chords essentially have their roles swapped from in meantone, as they now represent the [[Collection of chords|supermajor triad]] and [[Collection of chords|subminor triad]] respectively, and the minor chord is the more stable of the two. This can be seen by how the supermajor third is, in the 5-form, a flat fourth, serving a somewhat similar role to the diminished fifth in diatonic. The sus4 and sus2 triads are [[Collection of chords#Essentially tempered chords|essentially tempered chords]] here, stacking 4/3 twice to reach 7/4.

Due to existing in 2.3.7, Archy also supports the latal triads (bounded by a fourth, made from intervals near 250c, like 6:7:8), with 1/1-8/7-4/3 in particular appearing as part of the suspended tetrad.

{{cat|Temperaments}}

Archy

2026-01-27T22:09:28Z

Hkm:

'''Archy''', also called '''[[Perfect fifth#Tuning range|Superpyth]]''' (after its tuning range), is the temperament that tempers out the '''archytas comma,''' 64/63, equating [[2.3.7 subgroup|septal]] intervals with nearby [[Pythagorean tuning|diatonic]] ones. In Archy, the generator is a fourth, the period is an octave, and 2 flattened [[Perfect fourth|fourths]] of about 490 cents stack to a sharply tuned [[Septal subminor seventh|7th harmonic]] of about 980 cents. Equivalently, the pythagorean (9/8) major second is mapped to the same pitch as the septimal (8/7) major second.

Archy is usually tuned such that the subminor (7/6) third is close to accurately tuned; flatter tunings of the fourth lead to more accurate tunings of the 7th harmonic, at the cost of the usability of the diatonic scale. The tuning that leaves the harmonic seventh just places the perfect fourth at 484.4 cents, which leads to a diatonic scale with a small step at the uncomfortable size of 22 cents.

As a monocot temperament (a temperament generated by a perfect fourth or fifth), Archy can be notated with standard [[Diatonic notation|diatonic]] notation. However, this is somewhat awkward, as Archy is more cleanly analyzed as a 5-form temperament, producing an [[equipentatonic]] scale, so perhaps diamond-MOS or KISS notation with [[pentic]] would be better suited for it.

== Structural theory ==

=== Extensions ===
The following are extensions to prime 5 (i. e. ways to map intervals involving prime 5 onto the existing structure of 2.3.7 archy)

==== 5/4 as limma-flat major third (22 & 27), often called "Superpyth" ====
The canonical extension, equates 5/4 with the diatonic augmented second, or an octave-reduced stack of 9 fifths, which can be seen in the 5-form as a major third flattened by a diatonic semitone representing the [[septimal quartertone]] (36/35, the interval between 5/4 and 9/7) and the [[Meantone|syntonic comma]]. It can be seen as the 22 & 27 temperament. The preferred tuning range for the fifth in this extension tends to be somewhat flatter than that of archy; tunings where both the supermajor (9/7) and subminor (7/6) thirds are somewhat accurate are preferred.

==== 5/4 as doubly limma-flat major third (5 & 37) ====
This is an alternative extension, best tuned sharp of 27edo. Instead of flattening the major third by a diatonic semitone to reach the 5th harmonic, you flatten by two diatonic semitones. In diatonic notation, this means that 5/4 is the double-augmented unison.

== Compositional theory ==

=== Chords ===
In Archy, the diatonic major and minor chords essentially have their roles swapped from in meantone, as they now represent the [[Collection of chords|supermajor triad]] and [[Collection of chords|subminor triad]] respectively, and the minor chord is the more stable of the two. This can be seen by how the supermajor third is, in the 5-form, a flat fourth, serving a somewhat similar role to the diminished fifth in diatonic. The sus4 and sus2 triads are [[Collection of chords#Essentially tempered chords|essentially tempered chords]] here, stacking 4/3 twice to reach 7/4.

Due to existing in 2.3.7, Archy also supports the latal triads (bounded by a fourth, made from intervals near 250c, like 6:7:8), with 1/1-8/7-4/3 in particular appearing as part of the suspended tetrad.

{{cat|Temperaments}}

Glossary

2026-01-26T22:45:21Z

Hkm: /* Enharmonic */

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 or a 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 1
* [2, 3, 5, 15] is not a basis for the 5-limit, on account of not satisfying 2

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

Categories: RTT, JI, Math terms

== 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

== 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
== 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.
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 [https://xenreference.com/12edo 12edo], notes that are enharmonic in sense 1 are equated, which has led to a use of "enharmonic" to refer to other equivalences between notes of a scaleform in a particular tuning system (such as B# = Cb in [https://xenreference.com/19edo 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.

Categories: Core knowledge
== Extension ==
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 often 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. {{Adv|If you don't interpret the new intervals of a weak extension, the result is called ''contorsion''.}}

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 [[XenReference:Guidelines]].''

Categories: RTT, Somewhat technical

== 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 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'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the 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.

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 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 more commonly means prime-limit.

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 Xen 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
== Natave ==
The frequency ratio ''e''/1. Sometimes called a ''neper''. Mainly used in theoretical xen math.

== 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

== 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).
** 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 the equal division that supports it to be divisible by some number (namely, the number of 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. This is what makes a regular temperament "regular".

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 XenReference, ''scales are periodic unless stated otherwise.'' 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

== 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]]. 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 [[Mathematics of commas#Square superparticulars|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
== Union ==
The '''union''' of two scales/chords is a scale/chord with all pitches that occur in either scale/chord. In other words, it's a shorter way of saying "superimposition". If you change the offset between the two scales/chords, taking their union usually yields a different scale/chord.

Examples:
* A [[cross-set]] is a union of copies of the same scale placed on different offsets.
** 15edo pentawood uses two copies of 5edo offset by 1\15; 20edo pentawood uses two copies of 5edo offset by 1\20.
* ''Polysystemic tuning'' uses a union of multiple systems, for example 5edo and 7edo.

Categories: Math terms

== 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.

'''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.

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) or strict variety (SV). For example, [[MOS]] scales can be defined as scales that are MV2.

Categories: Scales

15edo

2026-01-26T22:11:47Z

Hkm: /* Non-diatonic scales */

[[File:HarmonicTable15.png|thumb|478x478px|The 2-dimensional harmonic table that serves as 15edo's defining feature.]]
'''15edo''', or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave [[2/1]]. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

== Theory ==

==== Edostep interpretations ====
15edo's edostep has the following interpretations in the 2...11 subgroup:

* 16/15 (the difference between 5/4 and 4/3)
* 25/24 (the difference between 5/4 and 6/5)
* 81/80 (the difference between 9/8 and 10/9)
* 36/35 (the difference between 5/4 and 9/7)
* 22/21 (the difference between 7/6 and 11/9)

==== JI approximation ====
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its [[Perfect fifth|fifth]] being 720 cents.
Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
{{Harmonics in ED|15|31|0}}
{| class="wikitable"
|+Thirds in 15edo
!Quality
|Minor
|Major
|-
!Cents
|320
|400
|-
!Just interpretation
|6/5
|5/4
|}

==== Chords ====
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

== Intervals ==

=== Interval categories ===
Here is a table of 15edo's intervals:
{| class="wikitable"
|+
!Name
!Degree
!Cents
!Approximate Ratios
|-
|Unison
|0
|0
|1/1
|-
|Minor second
|1
|80
|'''25/24,''' 16/15
|-
|Small major second
|2
|160
|'''10/9'''
|-
|Large major second
|3
|240
|'''8/7''', 9/8
|-
|Minor third
|4
|320
|'''6/5'''
|-
|Major third
|5
|400
|'''5/4'''
|-
|Perfect fourth
|6
|480
|'''4/3''', 21/16
|-
|Small tritone (acute fourth)
|7
|560
|'''11/8''', 7/5
|-
|Large tritone (grave fifth)
|8
|640
|'''16/11''', 10/7
|-
|Perfect fifth
|9
|720
|'''3/2''', 32/21
|-
|Minor sixth
|10
|800
|'''8/5'''
|-
|Major sixth
|11
|880
|'''5/3'''
|-
|Narrow minor seventh
|12
|960
|'''7/4''', 16/9
|-
|Wide minor seventh
|13
|1040
|'''9/5'''
|-
|Major seventh
|14
|1120
|'''48/25,''' 15/8
|-
|Octave
|15
|1200
|2/1
|}
=== The harmonic table ===
In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as "contorted" in 2.3.7 and 2.5. So, the only solution (other than stacking the harmonic 11) is to use a two-dimensional "circle", which forms a '''torus'''-shaped "harmonic table":
{| class="wikitable"
|+
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|}
We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.

Here,

* fifths are found by stepping 1 step up the Y-axis.
* harmonic sevenths are found by stepping 2 steps down the Y-axis.
* major thirds are found by stepping 1 step right along the X-axis.

Instead of two types of semitones, 15edo has four:

* the limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
* the classical chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
* the syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
* the diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts. Note that this semitone is actually the ''complement'' of the just diaschisma, since the diaschisma proper is a negative interval in 15edo.

== Scales ==
The Zarlino diatonic scale LmsLmLs or LsmLmLs is a way to preserve the classic major and minor dichotomies from 12edo. In the fourteen modes of these two scales, all thirds (i. e. intervals formable by ascending two scale-steps) are either major or minor, all fourths are either perfect fourths or large tritones, all fifths are perfect fifths or small tritones, and all sixths are either major or minor, with the following exceptions:

* in two of the fourteen modes, the fifth (i. e. interval formable by ascending four scale-steps) above the tonic is the 640c interval
* in two of the fourteen modes, the fourth is the 560c interval
* in two of the fourteen modes, the sixth is the 960c interval
* in two of the fourteen modes, the third is the 240c interval.

All of these exceptions involve intervals that can be better interpreted using intervals containing prime factors 7 and 11; thus, the Zarlino diatonic scale can bridge the classical 5-limit JI intervals with more complex intervals.

There is no MOS diatonic scale in 15edo, because a MOS diatonic scale is generated by stacking the fifth six times to get seven pitch classes, but in 15edo the process of stacking the fifth can only generate five pitch classes.

=== Non-diatonic scales ===
The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler|15|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler|15|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones. In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.

The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler|15|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler|15|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third span the same number of scale steps; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.

The scales generated by stacking either tritone are called pentic ({{Interval ruler|15|0, 80, 560, 640, 720, 1200}}), antidiatonic ({{Interval ruler|15|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler|15|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler|15|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

==== Blackwood ====
Blackwood is based on our "fifth-generated diatonic" {{{Interval ruler|15|0, 240, 480, 720, 960, 1200}}}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {{{Interval ruler|15|0, 160, 240, 400, 480, 640, 720, 880, 960, 1120, 1200}}}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of [[blackdye]], which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)

==== Augmented ====
Augmented is, conversely, based on major thirds. The basic augmented scale is {{{Interval ruler|15|0, 400, 800, 1200}}}, which is equivalent to 12edo's augmented triad. We can follow the same formula as blackwood and split this scale into large and small steps to get the following possible combinations:

- {{Interval ruler|15|0, 320, 400, 720, 800, 1120, 1200}} - the augmented scale proper, which contains fifths on exactly three of the six notes. 12edo contains a tuning of this scale, but we usually don't talk about it in 12edo.

- {{Interval ruler|15|0, 240, 400, 640, 800, 1040, 1200}} - the wholetone scale, which alternates the two sizes of wholetones and which contains many 7-limit intervals, but no fifths. This is similar to 12edo's wholetone scale, but since there are two different kinds of wholetones in 15edo, it uses both.

- {{Interval ruler|15|0, 160, 320, 400, 560, 720, 800, 960, 1120, 1200}} - the hyrulic scale, which is a way of combining the two prior scales to form a 9-note scale. This can be thought of as having perfect thirds, and major and minor fifths.

- {{Interval ruler|15|0, 240, 320, 400, 640, 720, 800, 1040, 1120, 1200}} - the tcherepnin scale, which is another combination of hard and soft triwood. This can also be thought of as having similar harmony to hyrulic, but with the minor fifth one step flatter.

=== MODMOS structures ===
Let's take the minor scale:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}} - minor

A common way to alter the minor scale is to sharpen the sixth and/or seventh degrees, resulting in the harmonic and melodic minor and variations of Dorian:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1120, 1200}} - harmonic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1120, 1200}} - dark melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1120, 1200}} - bright melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1040, 1200}}- "diatonyx" dorian

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1040, 1200}} - didymian dorian

A scale with a single generator (octave-equivalent, of course) is called a "MOS", and all these scales, including normal zarlino, are "MODMOSes" of '''onyx''', formed by sharping or flatting a few notes of it respectively. While technically, because this is an equal tuning, any 7-note scale can be a MODMOS of onyx, I try to reserve it for scales that have harmonic commonalities with diatonic or with onyx.

=== Constructing chords and splitting steps ===
A more interesting way to construct scales is to stack chords rather than single intervals. For example, if we stack 3 major chords, we get:

{{Interval ruler|15|0, 400, 720, 1200}} (major chord on root)

+ {{Interval ruler|15|240, 720, 1120}} (major chord on fifth)

+ {{Interval ruler|15|0, 480, 880, 1200}} (major chord descending from root)

= {{Interval ruler|15|0, 240, 400, 480, 720, 880, 1120, 1200}} - major

Surprise! Here's the zarlino scale again.

Let's try minor chords. This corresponds to flatting the seventh, third, and sixth degrees by one step.

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}}- minor

In fact, this is actually how the normal major and minor scales are constructed in diatonic, so it's neat to see that it checks out here.

So instead, let's try a different chord (let's say, for some reason, you cared a lot about wolf chords).

{{Interval ruler|15|0, 400, 640, 1200}} (wolf major chord on root)

+ {{Interval ruler|15|80, 640, 1040}} (wolf major chord on wolf fifth)

+ {{Interval ruler|15|0, 560, 960, 1200}} (wolf major chord descending from root)

= {{Interval ruler|15|0, 80, 400, 560, 640, 960, 1040, 1200}} - wolf major

You can call this the "wolf major scale" because of how it's constructed.

Let's try a kind of diminished chord, called a wolf diminished chord, made of two different sizes of minor third. Since the chord is a lot narrower, we'll need 4 chords instead of 3 to build a reasonable scale.

{{Interval ruler|15|0, 240, 560, 1200}} (wolf diminished chord on root)

+ {{Interval ruler|15|560, 800, 1120}} (wolf diminished chord on dim fifth)

+ {{Interval ruler|15|0, 640, 880, 1200}} (wolf diminished chord descending from root)

+ {{Interval ruler|15|80, 320, 640}} (wolf diminished chord descending from aug fourth)

= {{Interval ruler|15|0, 80, 240, 320, 560, 640, 800, 880, 1120, 1200}}- diminished

This scale also contains a "major wolf diminished" chord, which corresponds to the harmonic series sequence 5:6:7, much like a normal major chord corresponds to 4:5:6.

As can be seen, there's a lot of possibilities here.

Another way to build scales is by choosing a few key intervals, and splitting the jumps between them into steps. For example, we might decide that our key notes are {{{Interval ruler|15|0, 400, 480, 720, 1200}}}.

Here, we have two large jumps that can be split into steps, and there's a couple ways to do this, including ones that just result in zarlino and ones that result in scales we haven't seen before.

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1200}}- zaretan

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1200}} - legatus

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1040, 1200}}- decurion

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1040, 1200}} - kaiser

If we choose a different set of key intervals, we get a different set of possible scales:

{{Interval ruler|15|0, 240, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 400, 800, 640, 960, 1200}} - anhedonia

{{Interval ruler|15|0, 240, 400, 880, 720, 540, 960, 1200}} - mok

{{Interval ruler|15|0, 560, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 560, 400, 720, 960, 1200}}- amsel

{{Interval ruler|15|0, 320, 560, 400, 1120, 880, 960, 1200}} - drossel

=== Periodicity blocks ===
In 15edo, there are four primary ways of reaching the semitone on the harmonic table, corresponding to different copies of the semitone repeating periodically near the unison. It can be found as the difference between a stack of two fifths and a minor tone (the syntonic comma), the difference between a major third and a perfect fourth (the limma), the difference between the two tritones (the diaschisma), and the difference between a major and minor third (the classical chroma).

The standard zarlino scale can be defined by having the classical chroma and syntonic comma as its "chromas": if we create a tile that repeats at each multiple of these two chromas, the intervals within that tile are precisely those in the zarlino scale.

However, we can choose a different pair of semitones as our chromas, and get a different scale. Here's the classical chroma x diaschisma scale.

{{Interval ruler|15|0, 240, 320, 400, 560, 640, 800, 960, 1040, 1120, 1200}} - elena

{{Interval ruler|15|0, 240, 320, 400, 560, 720, 800, 960, 1040, 1120, 1200}} - kee'ra*

''*Note: in just intonation, the [[Collection of scales|kee'ra]] scale is a 7-limit "chair" scale, not a 5-limit periodicity block.''

There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.

And the classical chroma x limma scale:

{{Interval ruler|15|0, 320, 560, 800, 1040, 1200}}- myn

=== Regular temperaments ===
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo.

== Notation ==
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

{{Cat|Edos}}

15edo

2026-01-26T22:11:18Z

Hkm: /* Scales */

[[File:HarmonicTable15.png|thumb|478x478px|The 2-dimensional harmonic table that serves as 15edo's defining feature.]]
'''15edo''', or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave [[2/1]]. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

== Theory ==

==== Edostep interpretations ====
15edo's edostep has the following interpretations in the 2...11 subgroup:

* 16/15 (the difference between 5/4 and 4/3)
* 25/24 (the difference between 5/4 and 6/5)
* 81/80 (the difference between 9/8 and 10/9)
* 36/35 (the difference between 5/4 and 9/7)
* 22/21 (the difference between 7/6 and 11/9)

==== JI approximation ====
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its [[Perfect fifth|fifth]] being 720 cents.
Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
{{Harmonics in ED|15|31|0}}
{| class="wikitable"
|+Thirds in 15edo
!Quality
|Minor
|Major
|-
!Cents
|320
|400
|-
!Just interpretation
|6/5
|5/4
|}

==== Chords ====
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

== Intervals ==

=== Interval categories ===
Here is a table of 15edo's intervals:
{| class="wikitable"
|+
!Name
!Degree
!Cents
!Approximate Ratios
|-
|Unison
|0
|0
|1/1
|-
|Minor second
|1
|80
|'''25/24,''' 16/15
|-
|Small major second
|2
|160
|'''10/9'''
|-
|Large major second
|3
|240
|'''8/7''', 9/8
|-
|Minor third
|4
|320
|'''6/5'''
|-
|Major third
|5
|400
|'''5/4'''
|-
|Perfect fourth
|6
|480
|'''4/3''', 21/16
|-
|Small tritone (acute fourth)
|7
|560
|'''11/8''', 7/5
|-
|Large tritone (grave fifth)
|8
|640
|'''16/11''', 10/7
|-
|Perfect fifth
|9
|720
|'''3/2''', 32/21
|-
|Minor sixth
|10
|800
|'''8/5'''
|-
|Major sixth
|11
|880
|'''5/3'''
|-
|Narrow minor seventh
|12
|960
|'''7/4''', 16/9
|-
|Wide minor seventh
|13
|1040
|'''9/5'''
|-
|Major seventh
|14
|1120
|'''48/25,''' 15/8
|-
|Octave
|15
|1200
|2/1
|}
=== The harmonic table ===
In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as "contorted" in 2.3.7 and 2.5. So, the only solution (other than stacking the harmonic 11) is to use a two-dimensional "circle", which forms a '''torus'''-shaped "harmonic table":
{| class="wikitable"
|+
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|}
We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.

Here,

* fifths are found by stepping 1 step up the Y-axis.
* harmonic sevenths are found by stepping 2 steps down the Y-axis.
* major thirds are found by stepping 1 step right along the X-axis.

Instead of two types of semitones, 15edo has four:

* the limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
* the classical chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
* the syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
* the diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts. Note that this semitone is actually the ''complement'' of the just diaschisma, since the diaschisma proper is a negative interval in 15edo.

== Scales ==
The Zarlino diatonic scale LmsLmLs or LsmLmLs is a way to preserve the classic major and minor dichotomies from 12edo. In the fourteen modes of these two scales, all thirds (i. e. intervals formable by ascending two scale-steps) are either major or minor, all fourths are either perfect fourths or large tritones, all fifths are perfect fifths or small tritones, and all sixths are either major or minor, with the following exceptions:

* in two of the fourteen modes, the fifth (i. e. interval formable by ascending four scale-steps) above the tonic is the 640c interval
* in two of the fourteen modes, the fourth is the 560c interval
* in two of the fourteen modes, the sixth is the 960c interval
* in two of the fourteen modes, the third is the 240c interval.

All of these exceptions involve intervals that can be better interpreted using intervals containing prime factors 7 and 11; thus, the Zarlino diatonic scale can bridge the classical 5-limit JI intervals with more complex intervals.

There is no MOS diatonic scale in 15edo, because a MOS diatonic scale is generated by stacking the fifth six times to get seven pitch classes, but in 15edo the process of stacking the fifth can only generate five pitch classes.

=== Non-diatonic scales ===
The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler|15|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler|15|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones. In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.

The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler|15|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler|15|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 400-cent major third span the same number of scale steps; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.

The scales generated by stacking either tritone are called pentic ({{Interval ruler|15|0, 80, 560, 640, 720, 1200}}), peltonic ({{Interval ruler|15|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler|15|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler|15|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

==== Blackwood ====
Blackwood is based on our "fifth-generated diatonic" {{{Interval ruler|15|0, 240, 480, 720, 960, 1200}}}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {{{Interval ruler|15|0, 160, 240, 400, 480, 640, 720, 880, 960, 1120, 1200}}}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of [[blackdye]], which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)

==== Augmented ====
Augmented is, conversely, based on major thirds. The basic augmented scale is {{{Interval ruler|15|0, 400, 800, 1200}}}, which is equivalent to 12edo's augmented triad. We can follow the same formula as blackwood and split this scale into large and small steps to get the following possible combinations:

- {{Interval ruler|15|0, 320, 400, 720, 800, 1120, 1200}} - the augmented scale proper, which contains fifths on exactly three of the six notes. 12edo contains a tuning of this scale, but we usually don't talk about it in 12edo.

- {{Interval ruler|15|0, 240, 400, 640, 800, 1040, 1200}} - the wholetone scale, which alternates the two sizes of wholetones and which contains many 7-limit intervals, but no fifths. This is similar to 12edo's wholetone scale, but since there are two different kinds of wholetones in 15edo, it uses both.

- {{Interval ruler|15|0, 160, 320, 400, 560, 720, 800, 960, 1120, 1200}} - the hyrulic scale, which is a way of combining the two prior scales to form a 9-note scale. This can be thought of as having perfect thirds, and major and minor fifths.

- {{Interval ruler|15|0, 240, 320, 400, 640, 720, 800, 1040, 1120, 1200}} - the tcherepnin scale, which is another combination of hard and soft triwood. This can also be thought of as having similar harmony to hyrulic, but with the minor fifth one step flatter.

=== MODMOS structures ===
Let's take the minor scale:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}} - minor

A common way to alter the minor scale is to sharpen the sixth and/or seventh degrees, resulting in the harmonic and melodic minor and variations of Dorian:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1120, 1200}} - harmonic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1120, 1200}} - dark melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1120, 1200}} - bright melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1040, 1200}}- "diatonyx" dorian

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1040, 1200}} - didymian dorian

A scale with a single generator (octave-equivalent, of course) is called a "MOS", and all these scales, including normal zarlino, are "MODMOSes" of '''onyx''', formed by sharping or flatting a few notes of it respectively. While technically, because this is an equal tuning, any 7-note scale can be a MODMOS of onyx, I try to reserve it for scales that have harmonic commonalities with diatonic or with onyx.

=== Constructing chords and splitting steps ===
A more interesting way to construct scales is to stack chords rather than single intervals. For example, if we stack 3 major chords, we get:

{{Interval ruler|15|0, 400, 720, 1200}} (major chord on root)

+ {{Interval ruler|15|240, 720, 1120}} (major chord on fifth)

+ {{Interval ruler|15|0, 480, 880, 1200}} (major chord descending from root)

= {{Interval ruler|15|0, 240, 400, 480, 720, 880, 1120, 1200}} - major

Surprise! Here's the zarlino scale again.

Let's try minor chords. This corresponds to flatting the seventh, third, and sixth degrees by one step.

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}}- minor

In fact, this is actually how the normal major and minor scales are constructed in diatonic, so it's neat to see that it checks out here.

So instead, let's try a different chord (let's say, for some reason, you cared a lot about wolf chords).

{{Interval ruler|15|0, 400, 640, 1200}} (wolf major chord on root)

+ {{Interval ruler|15|80, 640, 1040}} (wolf major chord on wolf fifth)

+ {{Interval ruler|15|0, 560, 960, 1200}} (wolf major chord descending from root)

= {{Interval ruler|15|0, 80, 400, 560, 640, 960, 1040, 1200}} - wolf major

You can call this the "wolf major scale" because of how it's constructed.

Let's try a kind of diminished chord, called a wolf diminished chord, made of two different sizes of minor third. Since the chord is a lot narrower, we'll need 4 chords instead of 3 to build a reasonable scale.

{{Interval ruler|15|0, 240, 560, 1200}} (wolf diminished chord on root)

+ {{Interval ruler|15|560, 800, 1120}} (wolf diminished chord on dim fifth)

+ {{Interval ruler|15|0, 640, 880, 1200}} (wolf diminished chord descending from root)

+ {{Interval ruler|15|80, 320, 640}} (wolf diminished chord descending from aug fourth)

= {{Interval ruler|15|0, 80, 240, 320, 560, 640, 800, 880, 1120, 1200}}- diminished

This scale also contains a "major wolf diminished" chord, which corresponds to the harmonic series sequence 5:6:7, much like a normal major chord corresponds to 4:5:6.

As can be seen, there's a lot of possibilities here.

Another way to build scales is by choosing a few key intervals, and splitting the jumps between them into steps. For example, we might decide that our key notes are {{{Interval ruler|15|0, 400, 480, 720, 1200}}}.

Here, we have two large jumps that can be split into steps, and there's a couple ways to do this, including ones that just result in zarlino and ones that result in scales we haven't seen before.

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1200}}- zaretan

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1200}} - legatus

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1040, 1200}}- decurion

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1040, 1200}} - kaiser

If we choose a different set of key intervals, we get a different set of possible scales:

{{Interval ruler|15|0, 240, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 400, 800, 640, 960, 1200}} - anhedonia

{{Interval ruler|15|0, 240, 400, 880, 720, 540, 960, 1200}} - mok

{{Interval ruler|15|0, 560, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 560, 400, 720, 960, 1200}}- amsel

{{Interval ruler|15|0, 320, 560, 400, 1120, 880, 960, 1200}} - drossel

=== Periodicity blocks ===
In 15edo, there are four primary ways of reaching the semitone on the harmonic table, corresponding to different copies of the semitone repeating periodically near the unison. It can be found as the difference between a stack of two fifths and a minor tone (the syntonic comma), the difference between a major third and a perfect fourth (the limma), the difference between the two tritones (the diaschisma), and the difference between a major and minor third (the classical chroma).

The standard zarlino scale can be defined by having the classical chroma and syntonic comma as its "chromas": if we create a tile that repeats at each multiple of these two chromas, the intervals within that tile are precisely those in the zarlino scale.

However, we can choose a different pair of semitones as our chromas, and get a different scale. Here's the classical chroma x diaschisma scale.

{{Interval ruler|15|0, 240, 320, 400, 560, 640, 800, 960, 1040, 1120, 1200}} - elena

{{Interval ruler|15|0, 240, 320, 400, 560, 720, 800, 960, 1040, 1120, 1200}} - kee'ra*

''*Note: in just intonation, the [[Collection of scales|kee'ra]] scale is a 7-limit "chair" scale, not a 5-limit periodicity block.''

There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.

And the classical chroma x limma scale:

{{Interval ruler|15|0, 320, 560, 800, 1040, 1200}}- myn

=== Regular temperaments ===
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo.

== Notation ==
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

{{Cat|Edos}}

User:Hkm

2026-01-26T19:48:24Z

Hkm:

The following equivalence forms a definition of the opposites "intervallically otonal" and "intervallically utonal." If any one of the following statements is true, then they are all true; if any one is false, they are all false.

# JI interval c between notes A and B, where A is below B, is intervallically otonal
# A is intervallically utonal to B
# B is intervallically otonal to A
# c tends to have higher prime factors in the numerator and more prime factors in the denominator

If the JI interpretations of a cent value (from which that cent value draws its perceptual properties) tend to be otonal, then the cent value is called "roughly otonal." Even JI ratios that are "intervallically utonal" can be "roughly otonal": 32/25, which is quite close to 7/6 and is perceived as a tempered version of such, is an example. Simpler intervals tend to have more prominent effects and larger capture regions for rough otonality.

HKM finds that the perceptual root of a chord is the interval which tends to be the most roughly utonal and consonant to the other notes of the chord, and the perceptual tonic of a progression is (usually) determinable the same way. The strength of a resolution tends to approximate the strength of the root of the last chord as a perceptual tonic of the chord of the resolution and those chords preceding it.

User:Hkm

2026-01-26T19:10:59Z

Hkm:

The following equivalence forms a definition of the opposites "intervallically otonal" and "intervallically utonal." If any one of the following statements is true, then they are all true; if any one is false, they are all false.

# JI interval c between notes A and B, where A is below B, is intervallically otonal
# A is intervallically utonal to B
# B is intervallically otonal to A
# c tends to have higher prime factors in the numerator and more prime factors in the denominator

If the JI interpretations of a cent value (from which that cent value draws its perceptual properties) tend to be otonal, then the cent value is called "roughly otonal." Even JI ratios that are "intervallically utonal" can be "roughly otonal": 32/25, which is quite close to 7/6 and is perceived as a tempered version of such, is an example. Simpler intervals tend to have more prominent effects and larger capture regions for rough otonality.

HKM finds that the perceptual root of a chord is the interval which tends to be the most roughly utonal and consonant to the other notes of the chord, and the perceptual tonic of a progression is (usually) determinable the same way. The strength of a resolution tends to approximate the strengths of notes of the last chord as perceptual tonics of the chord of the resolution and those chords preceding it.

User:Hkm

2026-01-26T18:34:52Z

Hkm:

The following equivalence forms a definition of the opposites "intervallically otonal" and "intervallically utonal":

# JI interval c between notes A and B, where A is below B, is intervallically otonal
# A is intervallically utonal to B
# B is intervallically otonal to A
# c tends to have higher prime factors in the numerator and more prime factors in the denominator

If the JI interpretations of a cent value (from which that cent value draws its perceptual properties) tend to be otonal, then the cent value is called "roughly otonal." Even JI ratios that are "intervallically utonal" can be "roughly otonal": 32/25, which is quite close to 7/6 and is perceived as a tempered version of such, is an example. Simpler intervals tend to have more prominent effects and larger capture regions for rough otonality.

HKM finds that the perceptual root of a chord is the interval which tends to be the most roughly utonal and consonant to the other notes of the chord, and the perceptual tonic of a progression is (usually) determinable the same way. The strength of a resolution tends to approximate the strengths of notes of the last chord as perceptual tonics of the chord of the resolution and those chords preceding it.

Delta-rational chord

2026-01-26T17:48:43Z

Hkm:

A '''delta-rational''' ('''DR''') '''chord''' is a chord that has integer differences between harmonics, but the harmonics are not necessarily integers. That is, a chord is DR if it contains frequencies a, b, c, d, ... where one difference between two frequencies is a rational number times another difference between two frequencies. DR chords are typically described using the notation +a+b... showing ''relative'' frequency increments between adjacent chord notes, called the chord's '''delta signature'''. By ''relative'', it is meant that delta signatures are considered equivalent under scaling: +2+4+2 is the same delta signature as +1+2+1.

For example, the chord 0\13 – 3\13 – 8\13 – 924.159¢ is an exactly DR chord (with delta signature +1 +? +1), since the 3rd and 4th notes have exactly the same frequency difference as the interval 0–3\13. The actual 13edo chord 0 – 3 – 8 – 10\13 (0¢ – 277¢ – 738¢ – 923¢) is close to being delta-rational, because the frequency difference of the interval 8–10\13 is 0.994 times the frequency difference of the interval 0–3\13.

Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicings of JI chords stay JI.

Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple (i. e. low-number) delta signatures tend to be perceived as more concordant than other chords, even when the ratios between the notes themselves are not rational.

== Acoustics ==
The delta-rational acoustic effect is believed to result from synchronized interference beating occurring both among fundamental tones and among their lower harmonics. The strength of this effect varies based on factors such as register, timbre, and the complexity of the linear relationship involved. For instance, the effect tends to be weaker in chords with wider voicings, as well as in timbres where higher harmonics are more prominent thus obscuring the delta-rational relationships. The reason for focusing only on intervals between neighboring notes is that the tones within those intervals might psychoacoustically interfere with the beating patterns of the intervals themselves.

== Examples ==
# The chord 0c-400c-724.7c is a +1+1 chord (approximately 3.8473:4.8473:5.8473) and so is isodifferential (hence fully DR). It is close to the 15edo triad (0-5-9)\15 (0c-400c-720c).
# The chord 0c-281c-734.7c-923.6c is a +1+2+1 chord (approximately 5.675:6.675:8.675:9.675), and so is fully DR (but not isodifferential). It is close to the 13edo tetrad (0-3-8-10)\13 (0c-276.9c-738.5c-923.1c).
# The chord 0c-258.3c-771.7c-944.7c is a +1+?+1 chord (approximately 6.214:7.214:9.704:10.704), and thus a ''partially'' (not fully) DR tetrad. It is close to the 14edo tetrad (0-3-9-11)\14 (0c-257.1c-771.4c-942.9c).

== Definitions ==
* [[JI]] chords and chords that are subsets of isodifferential chords (these correspond to all chords of the form α : α + ''k''1 : ... : α + ''k''''n'' for any positive (possibly irrational) number α and integers ''k''1, ..., ''k''''n'') are special cases of delta-rational chords, but in these chords ''all'' intervals are rationally related in frequency space, which we call ''fully delta-rational''.
* If all notes are equally spaced in frequency, the chord is called ''isodifferential''.

Thus all isodifferential chords (including isoharmonic JI chords) are fully delta-rational, and all fully delta-rational chords (including all JI chords) are delta-rational.

Deltas that are ''free'', i.e. not required to be related to any other deltas are indicated with +?. For example, saying that a tetrad is "+1 +? +1" means the first two notes and the last two notes have equal frequency difference (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer intervals.

{{adv|If you have some sets of deltas related to each other but not to other sets of increments, you could write the related sets with variables a, b, ... or use one fewer letter by writing one set with positive integers without variables: a delta signature +a +b +a +b can also be written +1 +c +1 +c where c = b/a.}}

== See also ==
=== Expert ===
* [[Optimizing MOS scales for DR]]

=== Technical ===
* [[DR error measures]]

== External links ==
* [https://turbofishcrow.github.io/delta/ Inthar's DR chord explorer]
{{cat|Atypical ratios
Delta-rational harmony|*
}}

15edo

2026-01-26T17:42:54Z

Hkm: /* Non-diatonic scales */ Every 7-note scale is "analogous to diatonic" in the way that you're speaking of, and I don't know what you mean by "some kind of chord."

[[File:HarmonicTable15.png|thumb|478x478px|The 2-dimensional harmonic table that serves as 15edo's defining feature.]]
'''15edo''', or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave [[2/1]]. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

== Theory ==

==== Edostep interpretations ====
15edo's edostep has the following interpretations in the 2...11 subgroup:

* 16/15 (the difference between 5/4 and 4/3)
* 25/24 (the difference between 5/4 and 6/5)
* 81/80 (the difference between 9/8 and 10/9)
* 36/35 (the difference between 5/4 and 9/7)
* 22/21 (the difference between 7/6 and 11/9)

==== JI approximation ====
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its [[Perfect fifth|fifth]] being 720 cents.
Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
{{Harmonics in ED|15|31|0}}
{| class="wikitable"
|+Thirds in 15edo
!Quality
|Minor
|Major
|-
!Cents
|320
|400
|-
!Just interpretation
|6/5
|5/4
|}

==== Chords ====
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

== Intervals ==

=== Interval categories ===
Here is a table of 15edo's intervals:
{| class="wikitable"
|+
!Name
!Degree
!Cents
!Approximate Ratios
|-
|Unison
|0
|0
|1/1
|-
|Minor second
|1
|80
|'''25/24,''' 16/15
|-
|Small major second
|2
|160
|'''10/9'''
|-
|Large major second
|3
|240
|'''8/7''', 9/8
|-
|Minor third
|4
|320
|'''6/5'''
|-
|Major third
|5
|400
|'''5/4'''
|-
|Perfect fourth
|6
|480
|'''4/3''', 21/16
|-
|Small tritone (acute fourth)
|7
|560
|'''11/8''', 7/5
|-
|Large tritone (grave fifth)
|8
|640
|'''16/11''', 10/7
|-
|Perfect fifth
|9
|720
|'''3/2''', 32/21
|-
|Minor sixth
|10
|800
|'''8/5'''
|-
|Major sixth
|11
|880
|'''5/3'''
|-
|Narrow minor seventh
|12
|960
|'''7/4''', 16/9
|-
|Wide minor seventh
|13
|1040
|'''9/5'''
|-
|Major seventh
|14
|1120
|'''48/25,''' 15/8
|-
|Octave
|15
|1200
|2/1
|}
=== The harmonic table ===
In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as "contorted" in 2.3.7 and 2.5. So, the only solution (other than stacking the harmonic 11) is to use a two-dimensional "circle", which forms a '''torus'''-shaped "harmonic table":
{| class="wikitable"
|+
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|}
We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.

Here,

* fifths are found by stepping 1 step up the Y-axis.
* harmonic sevenths are found by stepping 2 steps down the Y-axis.
* major thirds are found by stepping 1 step right along the X-axis.

Instead of two types of semitones, 15edo has four:

* the limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
* the classical chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
* the syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
* the diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts. Note that this semitone is actually the ''complement'' of the just diaschisma, since the diaschisma proper is a negative interval in 15edo.

== Scales ==
The Zarlino diatonic scale LmsLmLs or LsmLmLs is a way to preserve the classic major and minor dichotomies from 12edo. In the fourteen modes of these two scales, all thirds (i. e. intervals formable by ascending two scale-steps) are either major or minor, all fourths are either perfect fourths or large tritones, all fifths are perfect fifths or small tritones, and all sixths are either major or minor, with the following exceptions:

* in two of the fourteen modes, the fifth (i. e. interval formable by ascending four scale-steps) above the tonic is the 640c interval
* in two of the fourteen modes, the fourth is the 560c interval
* in two of the fourteen modes, the sixth is the 960c interval
* in two of the fourteen modes, the third is the 240c interval.

All of these exceptions involve intervals that can be better interpreted using intervals containing prime factors 7 and 11; thus, the Zarlino diatonic scale can bridge the classical 5-limit JI intervals with more complex intervals.

There is no MOS diatonic scale in 15edo, because a MOS diatonic scale is generated by stacking the fifth six times to get seven pitch classes, but in 15edo the process of stacking the fifth can only generate five pitch classes.

=== Non-diatonic scales ===
The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler|15|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler|15|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones. In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.

The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler|15|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler|15|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 380-cent major third span the same number of scale steps; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.

The scales generated by stacking either tritone are called pentic ({{Interval ruler|15|0, 80, 560, 640, 720, 1200}}), peltonic ({{Interval ruler|15|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler|15|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler|15|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

==== Blackwood ====
Blackwood is based on our "fifth-generated diatonic" {{{Interval ruler|15|0, 240, 480, 720, 960, 1200}}}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {{{Interval ruler|15|0, 160, 240, 400, 480, 640, 720, 880, 960, 1120, 1200}}}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of [[blackdye]], which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)

==== Augmented ====
Augmented is, conversely, based on major thirds. The basic augmented scale is {{{Interval ruler|15|0, 400, 800, 1200}}}, which is equivalent to 12edo's augmented triad. We can follow the same formula as blackwood and split this scale into large and small steps to get the following possible combinations:

- {{Interval ruler|15|0, 320, 400, 720, 800, 1120, 1200}} - the augmented scale proper, which contains fifths on exactly three of the six notes. 12edo contains a tuning of this scale, but we usually don't talk about it in 12edo.

- {{Interval ruler|15|0, 240, 400, 640, 800, 1040, 1200}} - the wholetone scale, which alternates the two sizes of wholetones and which contains many 7-limit intervals, but no fifths. This is similar to 12edo's wholetone scale, but since there are two different kinds of wholetones in 15edo, it uses both.

- {{Interval ruler|15|0, 160, 320, 400, 560, 720, 800, 960, 1120, 1200}} - the hyrulic scale, which is a way of combining the two prior scales to form a 9-note scale. This can be thought of as having perfect thirds, and major and minor fifths.

- {{Interval ruler|15|0, 240, 320, 400, 640, 720, 800, 1040, 1120, 1200}} - the tcherepnin scale, which is another combination of hard and soft triwood. This can also be thought of as having similar harmony to hyrulic, but with the minor fifth one step flatter.

=== MODMOS structures ===
Let's take the minor scale:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}} - minor

A common way to alter the minor scale is to sharpen the sixth and/or seventh degrees, resulting in the harmonic and melodic minor and variations of Dorian:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1120, 1200}} - harmonic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1120, 1200}} - dark melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1120, 1200}} - bright melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1040, 1200}}- "diatonyx" dorian

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1040, 1200}} - didymian dorian

A scale with a single generator (octave-equivalent, of course) is called a "MOS", and all these scales, including normal zarlino, are "MODMOSes" of '''onyx''', formed by sharping or flatting a few notes of it respectively. While technically, because this is an equal tuning, any 7-note scale can be a MODMOS of onyx, I try to reserve it for scales that have harmonic commonalities with diatonic or with onyx.

=== Constructing chords and splitting steps ===
A more interesting way to construct scales is to stack chords rather than single intervals. For example, if we stack 3 major chords, we get:

{{Interval ruler|15|0, 400, 720, 1200}} (major chord on root)

+ {{Interval ruler|15|240, 720, 1120}} (major chord on fifth)

+ {{Interval ruler|15|0, 480, 880, 1200}} (major chord descending from root)

= {{Interval ruler|15|0, 240, 400, 480, 720, 880, 1120, 1200}} - major

Surprise! Here's the zarlino scale again.

Let's try minor chords. This corresponds to flatting the seventh, third, and sixth degrees by one step.

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}}- minor

In fact, this is actually how the normal major and minor scales are constructed in diatonic, so it's neat to see that it checks out here.

So instead, let's try a different chord (let's say, for some reason, you cared a lot about wolf chords).

{{Interval ruler|15|0, 400, 640, 1200}} (wolf major chord on root)

+ {{Interval ruler|15|80, 640, 1040}} (wolf major chord on wolf fifth)

+ {{Interval ruler|15|0, 560, 960, 1200}} (wolf major chord descending from root)

= {{Interval ruler|15|0, 80, 400, 560, 640, 960, 1040, 1200}} - wolf major

You can call this the "wolf major scale" because of how it's constructed.

Let's try a kind of diminished chord, called a wolf diminished chord, made of two different sizes of minor third. Since the chord is a lot narrower, we'll need 4 chords instead of 3 to build a reasonable scale.

{{Interval ruler|15|0, 240, 560, 1200}} (wolf diminished chord on root)

+ {{Interval ruler|15|560, 800, 1120}} (wolf diminished chord on dim fifth)

+ {{Interval ruler|15|0, 640, 880, 1200}} (wolf diminished chord descending from root)

+ {{Interval ruler|15|80, 320, 640}} (wolf diminished chord descending from aug fourth)

= {{Interval ruler|15|0, 80, 240, 320, 560, 640, 800, 880, 1120, 1200}}- diminished

This scale also contains a "major wolf diminished" chord, which corresponds to the harmonic series sequence 5:6:7, much like a normal major chord corresponds to 4:5:6.

As can be seen, there's a lot of possibilities here.

Another way to build scales is by choosing a few key intervals, and splitting the jumps between them into steps. For example, we might decide that our key notes are {{{Interval ruler|15|0, 400, 480, 720, 1200}}}.

Here, we have two large jumps that can be split into steps, and there's a couple ways to do this, including ones that just result in zarlino and ones that result in scales we haven't seen before.

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1200}}- zaretan

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1200}} - legatus

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1040, 1200}}- decurion

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1040, 1200}} - kaiser

If we choose a different set of key intervals, we get a different set of possible scales:

{{Interval ruler|15|0, 240, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 400, 800, 640, 960, 1200}} - anhedonia

{{Interval ruler|15|0, 240, 400, 880, 720, 540, 960, 1200}} - mok

{{Interval ruler|15|0, 560, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 560, 400, 720, 960, 1200}}- amsel

{{Interval ruler|15|0, 320, 560, 400, 1120, 880, 960, 1200}} - drossel

=== Periodicity blocks ===
In 15edo, there are four primary ways of reaching the semitone on the harmonic table, corresponding to different copies of the semitone repeating periodically near the unison. It can be found as the difference between a stack of two fifths and a minor tone (the syntonic comma), the difference between a major third and a perfect fourth (the limma), the difference between the two tritones (the diaschisma), and the difference between a major and minor third (the classical chroma).

The standard zarlino scale can be defined by having the classical chroma and syntonic comma as its "chromas": if we create a tile that repeats at each multiple of these two chromas, the intervals within that tile are precisely those in the zarlino scale.

However, we can choose a different pair of semitones as our chromas, and get a different scale. Here's the classical chroma x diaschisma scale.

{{Interval ruler|15|0, 240, 320, 400, 560, 640, 800, 960, 1040, 1120, 1200}} - elena

{{Interval ruler|15|0, 240, 320, 400, 560, 720, 800, 960, 1040, 1120, 1200}} - kee'ra*

''*Note: in just intonation, the [[Collection of scales|kee'ra]] scale is a 7-limit "chair" scale, not a 5-limit periodicity block.''

There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.

And the classical chroma x limma scale:

{{Interval ruler|15|0, 320, 560, 800, 1040, 1200}}- myn

=== Regular temperaments ===
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo.

== Notation ==
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

{{Cat|Edos}}

15edo

2026-01-26T17:42:22Z

Hkm: /* Non-diatonic scales */

[[File:HarmonicTable15.png|thumb|478x478px|The 2-dimensional harmonic table that serves as 15edo's defining feature.]]
'''15edo''', or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave [[2/1]]. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

== Theory ==

==== Edostep interpretations ====
15edo's edostep has the following interpretations in the 2...11 subgroup:

* 16/15 (the difference between 5/4 and 4/3)
* 25/24 (the difference between 5/4 and 6/5)
* 81/80 (the difference between 9/8 and 10/9)
* 36/35 (the difference between 5/4 and 9/7)
* 22/21 (the difference between 7/6 and 11/9)

==== JI approximation ====
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its [[Perfect fifth|fifth]] being 720 cents.
Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
{{Harmonics in ED|15|31|0}}
{| class="wikitable"
|+Thirds in 15edo
!Quality
|Minor
|Major
|-
!Cents
|320
|400
|-
!Just interpretation
|6/5
|5/4
|}

==== Chords ====
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

== Intervals ==

=== Interval categories ===
Here is a table of 15edo's intervals:
{| class="wikitable"
|+
!Name
!Degree
!Cents
!Approximate Ratios
|-
|Unison
|0
|0
|1/1
|-
|Minor second
|1
|80
|'''25/24,''' 16/15
|-
|Small major second
|2
|160
|'''10/9'''
|-
|Large major second
|3
|240
|'''8/7''', 9/8
|-
|Minor third
|4
|320
|'''6/5'''
|-
|Major third
|5
|400
|'''5/4'''
|-
|Perfect fourth
|6
|480
|'''4/3''', 21/16
|-
|Small tritone (acute fourth)
|7
|560
|'''11/8''', 7/5
|-
|Large tritone (grave fifth)
|8
|640
|'''16/11''', 10/7
|-
|Perfect fifth
|9
|720
|'''3/2''', 32/21
|-
|Minor sixth
|10
|800
|'''8/5'''
|-
|Major sixth
|11
|880
|'''5/3'''
|-
|Narrow minor seventh
|12
|960
|'''7/4''', 16/9
|-
|Wide minor seventh
|13
|1040
|'''9/5'''
|-
|Major seventh
|14
|1120
|'''48/25,''' 15/8
|-
|Octave
|15
|1200
|2/1
|}
=== The harmonic table ===
In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as "contorted" in 2.3.7 and 2.5. So, the only solution (other than stacking the harmonic 11) is to use a two-dimensional "circle", which forms a '''torus'''-shaped "harmonic table":
{| class="wikitable"
|+
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|}
We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.

Here,

* fifths are found by stepping 1 step up the Y-axis.
* harmonic sevenths are found by stepping 2 steps down the Y-axis.
* major thirds are found by stepping 1 step right along the X-axis.

Instead of two types of semitones, 15edo has four:

* the limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
* the classical chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
* the syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
* the diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts. Note that this semitone is actually the ''complement'' of the just diaschisma, since the diaschisma proper is a negative interval in 15edo.

== Scales ==
The Zarlino diatonic scale LmsLmLs or LsmLmLs is a way to preserve the classic major and minor dichotomies from 12edo. In the fourteen modes of these two scales, all thirds (i. e. intervals formable by ascending two scale-steps) are either major or minor, all fourths are either perfect fourths or large tritones, all fifths are perfect fifths or small tritones, and all sixths are either major or minor, with the following exceptions:

* in two of the fourteen modes, the fifth (i. e. interval formable by ascending four scale-steps) above the tonic is the 640c interval
* in two of the fourteen modes, the fourth is the 560c interval
* in two of the fourteen modes, the sixth is the 960c interval
* in two of the fourteen modes, the third is the 240c interval.

All of these exceptions involve intervals that can be better interpreted using intervals containing prime factors 7 and 11; thus, the Zarlino diatonic scale can bridge the classical 5-limit JI intervals with more complex intervals.

There is no MOS diatonic scale in 15edo, because a MOS diatonic scale is generated by stacking the fifth six times to get seven pitch classes, but in 15edo the process of stacking the fifth can only generate five pitch classes.

=== Non-diatonic scales ===
The scales generated by stacking the small major second, which are denoted "porcupine scales" due to their connection with the rank-2 [[regular temperament]] "porcupine", are onyx ({{Interval ruler|15|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler|15|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones. In onyx, four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale. The pine scale is simply the onyx scale with an extra note, which could function similarly to a "blue note" in blues.

The scales generated by stacking the 320-cent minor third are called smitonic ({{Interval ruler|15|0, 80, 320, 400, 640, 720, 960, 1200}}) and 4L 7s ({{Interval ruler|15|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, where a 320-cent minor third and 380-cent major third span the same number of scale steps; in general, smitonic contains many intervals close to JI ratios in the 5-limit and 7-limit.

The scales generated by stacking either tritone are called pentic ({{Interval ruler|15|0, 80, 560, 640, 720, 1200}}), peltonic ({{Interval ruler|15|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler|15|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and 2L 9s ({{Interval ruler|15|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

Surprisingly, the structure of peltonic is exactly analogous to that of 12edo's diatonic! It contains four perfect fifths, which correspond to 12edo's major sixths; 12edo's perfect fifths correspond to large tritones. Its extension, balzano, which Vector finds to be the most reasonable of the four tritone-generated scales to use, has six fifths, all of which can be used to build some kind of chord.

After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

==== Blackwood ====
Blackwood is based on our "fifth-generated diatonic" {{{Interval ruler|15|0, 240, 480, 720, 960, 1200}}}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {{{Interval ruler|15|0, 160, 240, 400, 480, 640, 720, 880, 960, 1120, 1200}}}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of [[blackdye]], which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)

==== Augmented ====
Augmented is, conversely, based on major thirds. The basic augmented scale is {{{Interval ruler|15|0, 400, 800, 1200}}}, which is equivalent to 12edo's augmented triad. We can follow the same formula as blackwood and split this scale into large and small steps to get the following possible combinations:

- {{Interval ruler|15|0, 320, 400, 720, 800, 1120, 1200}} - the augmented scale proper, which contains fifths on exactly three of the six notes. 12edo contains a tuning of this scale, but we usually don't talk about it in 12edo.

- {{Interval ruler|15|0, 240, 400, 640, 800, 1040, 1200}} - the wholetone scale, which alternates the two sizes of wholetones and which contains many 7-limit intervals, but no fifths. This is similar to 12edo's wholetone scale, but since there are two different kinds of wholetones in 15edo, it uses both.

- {{Interval ruler|15|0, 160, 320, 400, 560, 720, 800, 960, 1120, 1200}} - the hyrulic scale, which is a way of combining the two prior scales to form a 9-note scale. This can be thought of as having perfect thirds, and major and minor fifths.

- {{Interval ruler|15|0, 240, 320, 400, 640, 720, 800, 1040, 1120, 1200}} - the tcherepnin scale, which is another combination of hard and soft triwood. This can also be thought of as having similar harmony to hyrulic, but with the minor fifth one step flatter.

=== MODMOS structures ===
Let's take the minor scale:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}} - minor

A common way to alter the minor scale is to sharpen the sixth and/or seventh degrees, resulting in the harmonic and melodic minor and variations of Dorian:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1120, 1200}} - harmonic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1120, 1200}} - dark melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1120, 1200}} - bright melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1040, 1200}}- "diatonyx" dorian

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1040, 1200}} - didymian dorian

A scale with a single generator (octave-equivalent, of course) is called a "MOS", and all these scales, including normal zarlino, are "MODMOSes" of '''onyx''', formed by sharping or flatting a few notes of it respectively. While technically, because this is an equal tuning, any 7-note scale can be a MODMOS of onyx, I try to reserve it for scales that have harmonic commonalities with diatonic or with onyx.

=== Constructing chords and splitting steps ===
A more interesting way to construct scales is to stack chords rather than single intervals. For example, if we stack 3 major chords, we get:

{{Interval ruler|15|0, 400, 720, 1200}} (major chord on root)

+ {{Interval ruler|15|240, 720, 1120}} (major chord on fifth)

+ {{Interval ruler|15|0, 480, 880, 1200}} (major chord descending from root)

= {{Interval ruler|15|0, 240, 400, 480, 720, 880, 1120, 1200}} - major

Surprise! Here's the zarlino scale again.

Let's try minor chords. This corresponds to flatting the seventh, third, and sixth degrees by one step.

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}}- minor

In fact, this is actually how the normal major and minor scales are constructed in diatonic, so it's neat to see that it checks out here.

So instead, let's try a different chord (let's say, for some reason, you cared a lot about wolf chords).

{{Interval ruler|15|0, 400, 640, 1200}} (wolf major chord on root)

+ {{Interval ruler|15|80, 640, 1040}} (wolf major chord on wolf fifth)

+ {{Interval ruler|15|0, 560, 960, 1200}} (wolf major chord descending from root)

= {{Interval ruler|15|0, 80, 400, 560, 640, 960, 1040, 1200}} - wolf major

You can call this the "wolf major scale" because of how it's constructed.

Let's try a kind of diminished chord, called a wolf diminished chord, made of two different sizes of minor third. Since the chord is a lot narrower, we'll need 4 chords instead of 3 to build a reasonable scale.

{{Interval ruler|15|0, 240, 560, 1200}} (wolf diminished chord on root)

+ {{Interval ruler|15|560, 800, 1120}} (wolf diminished chord on dim fifth)

+ {{Interval ruler|15|0, 640, 880, 1200}} (wolf diminished chord descending from root)

+ {{Interval ruler|15|80, 320, 640}} (wolf diminished chord descending from aug fourth)

= {{Interval ruler|15|0, 80, 240, 320, 560, 640, 800, 880, 1120, 1200}}- diminished

This scale also contains a "major wolf diminished" chord, which corresponds to the harmonic series sequence 5:6:7, much like a normal major chord corresponds to 4:5:6.

As can be seen, there's a lot of possibilities here.

Another way to build scales is by choosing a few key intervals, and splitting the jumps between them into steps. For example, we might decide that our key notes are {{{Interval ruler|15|0, 400, 480, 720, 1200}}}.

Here, we have two large jumps that can be split into steps, and there's a couple ways to do this, including ones that just result in zarlino and ones that result in scales we haven't seen before.

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1200}}- zaretan

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1200}} - legatus

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1040, 1200}}- decurion

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1040, 1200}} - kaiser

If we choose a different set of key intervals, we get a different set of possible scales:

{{Interval ruler|15|0, 240, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 400, 800, 640, 960, 1200}} - anhedonia

{{Interval ruler|15|0, 240, 400, 880, 720, 540, 960, 1200}} - mok

{{Interval ruler|15|0, 560, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 560, 400, 720, 960, 1200}}- amsel

{{Interval ruler|15|0, 320, 560, 400, 1120, 880, 960, 1200}} - drossel

=== Periodicity blocks ===
In 15edo, there are four primary ways of reaching the semitone on the harmonic table, corresponding to different copies of the semitone repeating periodically near the unison. It can be found as the difference between a stack of two fifths and a minor tone (the syntonic comma), the difference between a major third and a perfect fourth (the limma), the difference between the two tritones (the diaschisma), and the difference between a major and minor third (the classical chroma).

The standard zarlino scale can be defined by having the classical chroma and syntonic comma as its "chromas": if we create a tile that repeats at each multiple of these two chromas, the intervals within that tile are precisely those in the zarlino scale.

However, we can choose a different pair of semitones as our chromas, and get a different scale. Here's the classical chroma x diaschisma scale.

{{Interval ruler|15|0, 240, 320, 400, 560, 640, 800, 960, 1040, 1120, 1200}} - elena

{{Interval ruler|15|0, 240, 320, 400, 560, 720, 800, 960, 1040, 1120, 1200}} - kee'ra*

''*Note: in just intonation, the [[Collection of scales|kee'ra]] scale is a 7-limit "chair" scale, not a 5-limit periodicity block.''

There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.

And the classical chroma x limma scale:

{{Interval ruler|15|0, 320, 560, 800, 1040, 1200}}- myn

=== Regular temperaments ===
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo.

== Notation ==
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

{{Cat|Edos}}

15edo

2026-01-26T17:35:01Z

Hkm: /* Non-diatonic scales */ ive never heard the chromatic scale called the "valentine scale" in my life

[[File:HarmonicTable15.png|thumb|478x478px|The 2-dimensional harmonic table that serves as 15edo's defining feature.]]
'''15edo''', or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave [[2/1]]. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

== Theory ==

==== Edostep interpretations ====
15edo's edostep has the following interpretations in the 2...11 subgroup:

* 16/15 (the difference between 5/4 and 4/3)
* 25/24 (the difference between 5/4 and 6/5)
* 81/80 (the difference between 9/8 and 10/9)
* 36/35 (the difference between 5/4 and 9/7)
* 22/21 (the difference between 7/6 and 11/9)

==== JI approximation ====
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its [[Perfect fifth|fifth]] being 720 cents.
Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
{{Harmonics in ED|15|31|0}}
{| class="wikitable"
|+Thirds in 15edo
!Quality
|Minor
|Major
|-
!Cents
|320
|400
|-
!Just interpretation
|6/5
|5/4
|}

==== Chords ====
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

== Intervals ==

=== Interval categories ===
Here is a table of 15edo's intervals:
{| class="wikitable"
|+
!Name
!Degree
!Cents
!Approximate Ratios
|-
|Unison
|0
|0
|1/1
|-
|Minor second
|1
|80
|'''25/24,''' 16/15
|-
|Small major second
|2
|160
|'''10/9'''
|-
|Large major second
|3
|240
|'''8/7''', 9/8
|-
|Minor third
|4
|320
|'''6/5'''
|-
|Major third
|5
|400
|'''5/4'''
|-
|Perfect fourth
|6
|480
|'''4/3''', 21/16
|-
|Small tritone (acute fourth)
|7
|560
|'''11/8''', 7/5
|-
|Large tritone (grave fifth)
|8
|640
|'''16/11''', 10/7
|-
|Perfect fifth
|9
|720
|'''3/2''', 32/21
|-
|Minor sixth
|10
|800
|'''8/5'''
|-
|Major sixth
|11
|880
|'''5/3'''
|-
|Narrow minor seventh
|12
|960
|'''7/4''', 16/9
|-
|Wide minor seventh
|13
|1040
|'''9/5'''
|-
|Major seventh
|14
|1120
|'''48/25,''' 15/8
|-
|Octave
|15
|1200
|2/1
|}
=== The harmonic table ===
In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as "contorted" in 2.3.7 and 2.5. So, the only solution (other than stacking the harmonic 11) is to use a two-dimensional "circle", which forms a '''torus'''-shaped "harmonic table":
{| class="wikitable"
|+
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|}
We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.

Here,

* fifths are found by stepping 1 step up the Y-axis.
* harmonic sevenths are found by stepping 2 steps down the Y-axis.
* major thirds are found by stepping 1 step right along the X-axis.

Instead of two types of semitones, 15edo has four:

* the limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
* the classical chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
* the syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
* the diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts. Note that this semitone is actually the ''complement'' of the just diaschisma, since the diaschisma proper is a negative interval in 15edo.

== Scales ==
The Zarlino diatonic scale LmsLmLs or LsmLmLs is a way to preserve the classic major and minor dichotomies from 12edo. In the fourteen modes of these two scales, all thirds (i. e. intervals formable by ascending two scale-steps) are either major or minor, all fourths are either perfect fourths or large tritones, all fifths are perfect fifths or small tritones, and all sixths are either major or minor, with the following exceptions:

* in two of the fourteen modes, the fifth (i. e. interval formable by ascending four scale-steps) above the tonic is the 640c interval
* in two of the fourteen modes, the fourth is the 560c interval
* in two of the fourteen modes, the sixth is the 960c interval
* in two of the fourteen modes, the third is the 240c interval.

All of these exceptions involve intervals that can be better interpreted using intervals containing prime factors 7 and 11; thus, the Zarlino diatonic scale can bridge the classical 5-limit JI intervals with more complex intervals.

There is no MOS diatonic scale in 15edo, because a MOS diatonic scale is generated by stacking the fifth six times to get seven pitch classes, but in 15edo the process of stacking the fifth can only generate five pitch classes.

=== Non-diatonic scales ===
Next, there are the porcupine scales: onyx ({{Interval ruler|15|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler|15|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones. In onyx (pine is really just a version of onyx with a "blue note"), four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale.

The third set of scales we'll look at is the kleismic scales, called smitonic ({{Interval ruler|15|0, 80, 320, 400, 640, 720, 960, 1200}}) and kleistonic ({{Interval ruler|15|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, with a major and minor third available on the same note; in general, smitonic contains many intervals of the 5-limit and 7-limit.

The fourth set of scales is the tritone-generated scales: pentic ({{Interval ruler|15|0, 80, 560, 640, 720, 1200}}), peltonic ({{Interval ruler|15|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler|15|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and semiquinary ({{Interval ruler|15|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

Surprisingly, the structure of peltonic is exactly analogous to that of 12edo's diatonic! It contains four perfect fifths, which correspond to 12edo's major sixths; 12edo's perfect fifths correspond to large tritones. Its extension, balzano, which I find the most reasonable of the four tritone-generated scales to use, has six fifths, all of which can be used to build some kind of chord.

After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

==== Blackwood ====
Blackwood is based on our "fifth-generated diatonic" {{{Interval ruler|15|0, 240, 480, 720, 960, 1200}}}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {{{Interval ruler|15|0, 160, 240, 400, 480, 640, 720, 880, 960, 1120, 1200}}}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of [[blackdye]], which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)

==== Augmented ====
Augmented is, conversely, based on major thirds. The basic augmented scale is {{{Interval ruler|15|0, 400, 800, 1200}}}, which is equivalent to 12edo's augmented triad. We can follow the same formula as blackwood and split this scale into large and small steps to get the following possible combinations:

- {{Interval ruler|15|0, 320, 400, 720, 800, 1120, 1200}} - the augmented scale proper, which contains fifths on exactly three of the six notes. 12edo contains a tuning of this scale, but we usually don't talk about it in 12edo.

- {{Interval ruler|15|0, 240, 400, 640, 800, 1040, 1200}} - the wholetone scale, which alternates the two sizes of wholetones and which contains many 7-limit intervals, but no fifths. This is similar to 12edo's wholetone scale, but since there are two different kinds of wholetones in 15edo, it uses both.

- {{Interval ruler|15|0, 160, 320, 400, 560, 720, 800, 960, 1120, 1200}} - the hyrulic scale, which is a way of combining the two prior scales to form a 9-note scale. This can be thought of as having perfect thirds, and major and minor fifths.

- {{Interval ruler|15|0, 240, 320, 400, 640, 720, 800, 1040, 1120, 1200}} - the tcherepnin scale, which is another combination of hard and soft triwood. This can also be thought of as having similar harmony to hyrulic, but with the minor fifth one step flatter.

=== MODMOS structures ===
Let's take the minor scale:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}} - minor

A common way to alter the minor scale is to sharpen the sixth and/or seventh degrees, resulting in the harmonic and melodic minor and variations of Dorian:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1120, 1200}} - harmonic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1120, 1200}} - dark melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1120, 1200}} - bright melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1040, 1200}}- "diatonyx" dorian

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1040, 1200}} - didymian dorian

A scale with a single generator (octave-equivalent, of course) is called a "MOS", and all these scales, including normal zarlino, are "MODMOSes" of '''onyx''', formed by sharping or flatting a few notes of it respectively. While technically, because this is an equal tuning, any 7-note scale can be a MODMOS of onyx, I try to reserve it for scales that have harmonic commonalities with diatonic or with onyx.

=== Constructing chords and splitting steps ===
A more interesting way to construct scales is to stack chords rather than single intervals. For example, if we stack 3 major chords, we get:

{{Interval ruler|15|0, 400, 720, 1200}} (major chord on root)

+ {{Interval ruler|15|240, 720, 1120}} (major chord on fifth)

+ {{Interval ruler|15|0, 480, 880, 1200}} (major chord descending from root)

= {{Interval ruler|15|0, 240, 400, 480, 720, 880, 1120, 1200}} - major

Surprise! Here's the zarlino scale again.

Let's try minor chords. This corresponds to flatting the seventh, third, and sixth degrees by one step.

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}}- minor

In fact, this is actually how the normal major and minor scales are constructed in diatonic, so it's neat to see that it checks out here.

So instead, let's try a different chord (let's say, for some reason, you cared a lot about wolf chords).

{{Interval ruler|15|0, 400, 640, 1200}} (wolf major chord on root)

+ {{Interval ruler|15|80, 640, 1040}} (wolf major chord on wolf fifth)

+ {{Interval ruler|15|0, 560, 960, 1200}} (wolf major chord descending from root)

= {{Interval ruler|15|0, 80, 400, 560, 640, 960, 1040, 1200}} - wolf major

You can call this the "wolf major scale" because of how it's constructed.

Let's try a kind of diminished chord, called a wolf diminished chord, made of two different sizes of minor third. Since the chord is a lot narrower, we'll need 4 chords instead of 3 to build a reasonable scale.

{{Interval ruler|15|0, 240, 560, 1200}} (wolf diminished chord on root)

+ {{Interval ruler|15|560, 800, 1120}} (wolf diminished chord on dim fifth)

+ {{Interval ruler|15|0, 640, 880, 1200}} (wolf diminished chord descending from root)

+ {{Interval ruler|15|80, 320, 640}} (wolf diminished chord descending from aug fourth)

= {{Interval ruler|15|0, 80, 240, 320, 560, 640, 800, 880, 1120, 1200}}- diminished

This scale also contains a "major wolf diminished" chord, which corresponds to the harmonic series sequence 5:6:7, much like a normal major chord corresponds to 4:5:6.

As can be seen, there's a lot of possibilities here.

Another way to build scales is by choosing a few key intervals, and splitting the jumps between them into steps. For example, we might decide that our key notes are {{{Interval ruler|15|0, 400, 480, 720, 1200}}}.

Here, we have two large jumps that can be split into steps, and there's a couple ways to do this, including ones that just result in zarlino and ones that result in scales we haven't seen before.

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1200}}- zaretan

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1200}} - legatus

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1040, 1200}}- decurion

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1040, 1200}} - kaiser

If we choose a different set of key intervals, we get a different set of possible scales:

{{Interval ruler|15|0, 240, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 400, 800, 640, 960, 1200}} - anhedonia

{{Interval ruler|15|0, 240, 400, 880, 720, 540, 960, 1200}} - mok

{{Interval ruler|15|0, 560, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 560, 400, 720, 960, 1200}}- amsel

{{Interval ruler|15|0, 320, 560, 400, 1120, 880, 960, 1200}} - drossel

=== Periodicity blocks ===
In 15edo, there are four primary ways of reaching the semitone on the harmonic table, corresponding to different copies of the semitone repeating periodically near the unison. It can be found as the difference between a stack of two fifths and a minor tone (the syntonic comma), the difference between a major third and a perfect fourth (the limma), the difference between the two tritones (the diaschisma), and the difference between a major and minor third (the classical chroma).

The standard zarlino scale can be defined by having the classical chroma and syntonic comma as its "chromas": if we create a tile that repeats at each multiple of these two chromas, the intervals within that tile are precisely those in the zarlino scale.

However, we can choose a different pair of semitones as our chromas, and get a different scale. Here's the classical chroma x diaschisma scale.

{{Interval ruler|15|0, 240, 320, 400, 560, 640, 800, 960, 1040, 1120, 1200}} - elena

{{Interval ruler|15|0, 240, 320, 400, 560, 720, 800, 960, 1040, 1120, 1200}} - kee'ra*

''*Note: in just intonation, the [[Collection of scales|kee'ra]] scale is a 7-limit "chair" scale, not a 5-limit periodicity block.''

There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.

And the classical chroma x limma scale:

{{Interval ruler|15|0, 320, 560, 800, 1040, 1200}}- myn

=== Regular temperaments ===
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo.

== Notation ==
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

{{Cat|Edos}}

15edo

2026-01-26T17:33:18Z

Hkm: /* Scales */ larger edos give you more scales. there's nothing special about 15 in how many scales it gives you access to

[[File:HarmonicTable15.png|thumb|478x478px|The 2-dimensional harmonic table that serves as 15edo's defining feature.]]
'''15edo''', or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave [[2/1]]. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

== Theory ==

==== Edostep interpretations ====
15edo's edostep has the following interpretations in the 2...11 subgroup:

* 16/15 (the difference between 5/4 and 4/3)
* 25/24 (the difference between 5/4 and 6/5)
* 81/80 (the difference between 9/8 and 10/9)
* 36/35 (the difference between 5/4 and 9/7)
* 22/21 (the difference between 7/6 and 11/9)

==== JI approximation ====
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its [[Perfect fifth|fifth]] being 720 cents.
Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
{{Harmonics in ED|15|31|0}}
{| class="wikitable"
|+Thirds in 15edo
!Quality
|Minor
|Major
|-
!Cents
|320
|400
|-
!Just interpretation
|6/5
|5/4
|}

==== Chords ====
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

== Intervals ==

=== Interval categories ===
Here is a table of 15edo's intervals:
{| class="wikitable"
|+
!Name
!Degree
!Cents
!Approximate Ratios
|-
|Unison
|0
|0
|1/1
|-
|Minor second
|1
|80
|'''25/24,''' 16/15
|-
|Small major second
|2
|160
|'''10/9'''
|-
|Large major second
|3
|240
|'''8/7''', 9/8
|-
|Minor third
|4
|320
|'''6/5'''
|-
|Major third
|5
|400
|'''5/4'''
|-
|Perfect fourth
|6
|480
|'''4/3''', 21/16
|-
|Small tritone (acute fourth)
|7
|560
|'''11/8''', 7/5
|-
|Large tritone (grave fifth)
|8
|640
|'''16/11''', 10/7
|-
|Perfect fifth
|9
|720
|'''3/2''', 32/21
|-
|Minor sixth
|10
|800
|'''8/5'''
|-
|Major sixth
|11
|880
|'''5/3'''
|-
|Narrow minor seventh
|12
|960
|'''7/4''', 16/9
|-
|Wide minor seventh
|13
|1040
|'''9/5'''
|-
|Major seventh
|14
|1120
|'''48/25,''' 15/8
|-
|Octave
|15
|1200
|2/1
|}
=== The harmonic table ===
In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as "contorted" in 2.3.7 and 2.5. So, the only solution (other than stacking the harmonic 11) is to use a two-dimensional "circle", which forms a '''torus'''-shaped "harmonic table":
{| class="wikitable"
|+
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|}
We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.

Here,

* fifths are found by stepping 1 step up the Y-axis.
* harmonic sevenths are found by stepping 2 steps down the Y-axis.
* major thirds are found by stepping 1 step right along the X-axis.

Instead of two types of semitones, 15edo has four:

* the limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
* the classical chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
* the syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
* the diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts. Note that this semitone is actually the ''complement'' of the just diaschisma, since the diaschisma proper is a negative interval in 15edo.

== Scales ==
The Zarlino diatonic scale LmsLmLs or LsmLmLs is a way to preserve the classic major and minor dichotomies from 12edo. In the fourteen modes of these two scales, all thirds (i. e. intervals formable by ascending two scale-steps) are either major or minor, all fourths are either perfect fourths or large tritones, all fifths are perfect fifths or small tritones, and all sixths are either major or minor, with the following exceptions:

* in two of the fourteen modes, the fifth (i. e. interval formable by ascending four scale-steps) above the tonic is the 640c interval
* in two of the fourteen modes, the fourth is the 560c interval
* in two of the fourteen modes, the sixth is the 960c interval
* in two of the fourteen modes, the third is the 240c interval.

All of these exceptions involve intervals that can be better interpreted using intervals containing prime factors 7 and 11; thus, the Zarlino diatonic scale can bridge the classical 5-limit JI intervals with more complex intervals.

There is no MOS diatonic scale in 15edo, because a MOS diatonic scale is generated by stacking the fifth six times to get seven pitch classes, but in 15edo the process of stacking the fifth can only generate five pitch classes.

=== Non-diatonic scales ===
We first start with 15edo's chromatic scale. In 15edo, the chromatic scale is also called the "[[Carlos Alpha|valentine]] scale": 3 chromatic steps equals a major tone, 5 is a major third, and 9 is a perfect fifth. Being the chromatic scale, valentine contains all 15 notes: {{{Interval ruler|15|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}}

Next, there are the porcupine scales: onyx ({{Interval ruler|15|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler|15|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones. In onyx (pine is really just a version of onyx with a "blue note"), four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale.

The third set of scales we'll look at is the kleismic scales, called smitonic ({{Interval ruler|15|0, 80, 320, 400, 640, 720, 960, 1200}}) and kleistonic ({{Interval ruler|15|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, with a major and minor third available on the same note; in general, smitonic contains many intervals of the 5-limit and 7-limit.

The fourth set of scales is the tritone-generated scales: pentic ({{Interval ruler|15|0, 80, 560, 640, 720, 1200}}), peltonic ({{Interval ruler|15|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler|15|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and semiquinary ({{Interval ruler|15|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

Surprisingly, the structure of peltonic is exactly analogous to that of 12edo's diatonic! It contains four perfect fifths, which correspond to 12edo's major sixths; 12edo's perfect fifths correspond to large tritones. Its extension, balzano, which I find the most reasonable of the four tritone-generated scales to use, has six fifths, all of which can be used to build some kind of chord.

After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

==== Blackwood ====
Blackwood is based on our "fifth-generated diatonic" {{{Interval ruler|15|0, 240, 480, 720, 960, 1200}}}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {{{Interval ruler|15|0, 160, 240, 400, 480, 640, 720, 880, 960, 1120, 1200}}}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of [[blackdye]], which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)

==== Augmented ====
Augmented is, conversely, based on major thirds. The basic augmented scale is {{{Interval ruler|15|0, 400, 800, 1200}}}, which is equivalent to 12edo's augmented triad. We can follow the same formula as blackwood and split this scale into large and small steps to get the following possible combinations:

- {{Interval ruler|15|0, 320, 400, 720, 800, 1120, 1200}} - the augmented scale proper, which contains fifths on exactly three of the six notes. 12edo contains a tuning of this scale, but we usually don't talk about it in 12edo.

- {{Interval ruler|15|0, 240, 400, 640, 800, 1040, 1200}} - the wholetone scale, which alternates the two sizes of wholetones and which contains many 7-limit intervals, but no fifths. This is similar to 12edo's wholetone scale, but since there are two different kinds of wholetones in 15edo, it uses both.

- {{Interval ruler|15|0, 160, 320, 400, 560, 720, 800, 960, 1120, 1200}} - the hyrulic scale, which is a way of combining the two prior scales to form a 9-note scale. This can be thought of as having perfect thirds, and major and minor fifths.

- {{Interval ruler|15|0, 240, 320, 400, 640, 720, 800, 1040, 1120, 1200}} - the tcherepnin scale, which is another combination of hard and soft triwood. This can also be thought of as having similar harmony to hyrulic, but with the minor fifth one step flatter.

=== MODMOS structures ===
Let's take the minor scale:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}} - minor

A common way to alter the minor scale is to sharpen the sixth and/or seventh degrees, resulting in the harmonic and melodic minor and variations of Dorian:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1120, 1200}} - harmonic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1120, 1200}} - dark melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1120, 1200}} - bright melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1040, 1200}}- "diatonyx" dorian

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1040, 1200}} - didymian dorian

A scale with a single generator (octave-equivalent, of course) is called a "MOS", and all these scales, including normal zarlino, are "MODMOSes" of '''onyx''', formed by sharping or flatting a few notes of it respectively. While technically, because this is an equal tuning, any 7-note scale can be a MODMOS of onyx, I try to reserve it for scales that have harmonic commonalities with diatonic or with onyx.

=== Constructing chords and splitting steps ===
A more interesting way to construct scales is to stack chords rather than single intervals. For example, if we stack 3 major chords, we get:

{{Interval ruler|15|0, 400, 720, 1200}} (major chord on root)

+ {{Interval ruler|15|240, 720, 1120}} (major chord on fifth)

+ {{Interval ruler|15|0, 480, 880, 1200}} (major chord descending from root)

= {{Interval ruler|15|0, 240, 400, 480, 720, 880, 1120, 1200}} - major

Surprise! Here's the zarlino scale again.

Let's try minor chords. This corresponds to flatting the seventh, third, and sixth degrees by one step.

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}}- minor

In fact, this is actually how the normal major and minor scales are constructed in diatonic, so it's neat to see that it checks out here.

So instead, let's try a different chord (let's say, for some reason, you cared a lot about wolf chords).

{{Interval ruler|15|0, 400, 640, 1200}} (wolf major chord on root)

+ {{Interval ruler|15|80, 640, 1040}} (wolf major chord on wolf fifth)

+ {{Interval ruler|15|0, 560, 960, 1200}} (wolf major chord descending from root)

= {{Interval ruler|15|0, 80, 400, 560, 640, 960, 1040, 1200}} - wolf major

You can call this the "wolf major scale" because of how it's constructed.

Let's try a kind of diminished chord, called a wolf diminished chord, made of two different sizes of minor third. Since the chord is a lot narrower, we'll need 4 chords instead of 3 to build a reasonable scale.

{{Interval ruler|15|0, 240, 560, 1200}} (wolf diminished chord on root)

+ {{Interval ruler|15|560, 800, 1120}} (wolf diminished chord on dim fifth)

+ {{Interval ruler|15|0, 640, 880, 1200}} (wolf diminished chord descending from root)

+ {{Interval ruler|15|80, 320, 640}} (wolf diminished chord descending from aug fourth)

= {{Interval ruler|15|0, 80, 240, 320, 560, 640, 800, 880, 1120, 1200}}- diminished

This scale also contains a "major wolf diminished" chord, which corresponds to the harmonic series sequence 5:6:7, much like a normal major chord corresponds to 4:5:6.

As can be seen, there's a lot of possibilities here.

Another way to build scales is by choosing a few key intervals, and splitting the jumps between them into steps. For example, we might decide that our key notes are {{{Interval ruler|15|0, 400, 480, 720, 1200}}}.

Here, we have two large jumps that can be split into steps, and there's a couple ways to do this, including ones that just result in zarlino and ones that result in scales we haven't seen before.

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1200}}- zaretan

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1200}} - legatus

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1040, 1200}}- decurion

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1040, 1200}} - kaiser

If we choose a different set of key intervals, we get a different set of possible scales:

{{Interval ruler|15|0, 240, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 400, 800, 640, 960, 1200}} - anhedonia

{{Interval ruler|15|0, 240, 400, 880, 720, 540, 960, 1200}} - mok

{{Interval ruler|15|0, 560, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 560, 400, 720, 960, 1200}}- amsel

{{Interval ruler|15|0, 320, 560, 400, 1120, 880, 960, 1200}} - drossel

=== Periodicity blocks ===
In 15edo, there are four primary ways of reaching the semitone on the harmonic table, corresponding to different copies of the semitone repeating periodically near the unison. It can be found as the difference between a stack of two fifths and a minor tone (the syntonic comma), the difference between a major third and a perfect fourth (the limma), the difference between the two tritones (the diaschisma), and the difference between a major and minor third (the classical chroma).

The standard zarlino scale can be defined by having the classical chroma and syntonic comma as its "chromas": if we create a tile that repeats at each multiple of these two chromas, the intervals within that tile are precisely those in the zarlino scale.

However, we can choose a different pair of semitones as our chromas, and get a different scale. Here's the classical chroma x diaschisma scale.

{{Interval ruler|15|0, 240, 320, 400, 560, 640, 800, 960, 1040, 1120, 1200}} - elena

{{Interval ruler|15|0, 240, 320, 400, 560, 720, 800, 960, 1040, 1120, 1200}} - kee'ra*

''*Note: in just intonation, the [[Collection of scales|kee'ra]] scale is a 7-limit "chair" scale, not a 5-limit periodicity block.''

There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.

And the classical chroma x limma scale:

{{Interval ruler|15|0, 320, 560, 800, 1040, 1200}}- myn

=== Regular temperaments ===
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo.

== Notation ==
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

{{Cat|Edos}}

15edo

2026-01-26T17:32:27Z

Hkm: /* Interval categories */

[[File:HarmonicTable15.png|thumb|478x478px|The 2-dimensional harmonic table that serves as 15edo's defining feature.]]
'''15edo''', or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave [[2/1]]. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

== Theory ==

==== Edostep interpretations ====
15edo's edostep has the following interpretations in the 2...11 subgroup:

* 16/15 (the difference between 5/4 and 4/3)
* 25/24 (the difference between 5/4 and 6/5)
* 81/80 (the difference between 9/8 and 10/9)
* 36/35 (the difference between 5/4 and 9/7)
* 22/21 (the difference between 7/6 and 11/9)

==== JI approximation ====
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its [[Perfect fifth|fifth]] being 720 cents.
Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
{{Harmonics in ED|15|31|0}}
{| class="wikitable"
|+Thirds in 15edo
!Quality
|Minor
|Major
|-
!Cents
|320
|400
|-
!Just interpretation
|6/5
|5/4
|}

==== Chords ====
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

== Intervals ==

=== Interval categories ===
Here is a table of 15edo's intervals:
{| class="wikitable"
|+
!Name
!Degree
!Cents
!Approximate Ratios
|-
|Unison
|0
|0
|1/1
|-
|Minor second
|1
|80
|'''25/24,''' 16/15
|-
|Small major second
|2
|160
|'''10/9'''
|-
|Large major second
|3
|240
|'''8/7''', 9/8
|-
|Minor third
|4
|320
|'''6/5'''
|-
|Major third
|5
|400
|'''5/4'''
|-
|Perfect fourth
|6
|480
|'''4/3''', 21/16
|-
|Small tritone (acute fourth)
|7
|560
|'''11/8''', 7/5
|-
|Large tritone (grave fifth)
|8
|640
|'''16/11''', 10/7
|-
|Perfect fifth
|9
|720
|'''3/2''', 32/21
|-
|Minor sixth
|10
|800
|'''8/5'''
|-
|Major sixth
|11
|880
|'''5/3'''
|-
|Narrow minor seventh
|12
|960
|'''7/4''', 16/9
|-
|Wide minor seventh
|13
|1040
|'''9/5'''
|-
|Major seventh
|14
|1120
|'''48/25,''' 15/8
|-
|Octave
|15
|1200
|2/1
|}
=== The harmonic table ===
In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as "contorted" in 2.3.7 and 2.5. So, the only solution (other than stacking the harmonic 11) is to use a two-dimensional "circle", which forms a '''torus'''-shaped "harmonic table":
{| class="wikitable"
|+
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|-
|480c
|880c
|80c
|480c
|880c
|80c
|480c
|-
|960c
|160c
|560c
|960c
|160c
|560c
|960c
|-
|240c
|640c
|1040c
|240c
|640c
|1040c
|240c
|-
|720c
|1120c
|320c
|720c
|1120c
|320c
|720c
|-
|'''0c'''
|400c
|800c
|'''0c'''
|400c
|800c
|'''0c'''
|}
We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.

Here,

* fifths are found by stepping 1 step up the Y-axis.
* harmonic sevenths are found by stepping 2 steps down the Y-axis.
* major thirds are found by stepping 1 step right along the X-axis.

Instead of two types of semitones, 15edo has four:

* the limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
* the classical chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
* the syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
* the diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts. Note that this semitone is actually the ''complement'' of the just diaschisma, since the diaschisma proper is a negative interval in 15edo.

== Scales ==
One notable thing about 15edo is how many different kinds of scales it gives you access to.

The Zarlino diatonic scale LmsLmLs or LsmLmLs is a way to preserve the classic major and minor dichotomies from 12edo. In the fourteen modes of these two scales, all thirds (i. e. intervals formable by ascending two scale-steps) are either major or minor, all fourths are either perfect fourths or large tritones, all fifths are perfect fifths or small tritones, and all sixths are either major or minor, with the following exceptions:

* in two of the fourteen modes, the fifth (i. e. interval formable by ascending four scale-steps) above the tonic is the 640c interval
* in two of the fourteen modes, the fourth is the 560c interval
* in two of the fourteen modes, the sixth is the 960c interval
* in two of the fourteen modes, the third is the 240c interval.

All of these exceptions involve intervals that can be better interpreted using intervals containing prime factors 7 and 11; thus, the Zarlino diatonic scale can bridge the classical 5-limit JI intervals with more complex intervals.

There is no MOS diatonic scale in 15edo, because a MOS diatonic scale is generated by stacking the fifth six times to get seven pitch classes, but in 15edo the process of stacking the fifth can only generate five pitch classes.

=== Non-diatonic scales ===
We first start with 15edo's chromatic scale. In 15edo, the chromatic scale is also called the "[[Carlos Alpha|valentine]] scale": 3 chromatic steps equals a major tone, 5 is a major third, and 9 is a perfect fifth. Being the chromatic scale, valentine contains all 15 notes: {{{Interval ruler|15|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}}

Next, there are the porcupine scales: onyx ({{Interval ruler|15|0, 160, 320, 480, 720, 880, 1040, 1200}}) and pine ({{Interval ruler|15|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}). The steps of the porcupine scales are minor tones. In onyx (pine is really just a version of onyx with a "blue note"), four notes have perfect fifths above them, and four notes have perfect fourths, overlapping at one note which has both. Onyx has a special relationship to the diatonic scale, however, and this can be found by sharping the third and seventh degrees of onyx to produce the zarlino diatonic scale.

The third set of scales we'll look at is the kleismic scales, called smitonic ({{Interval ruler|15|0, 80, 320, 400, 640, 720, 960, 1200}}) and kleistonic ({{Interval ruler|15|0, 80, 160, 320, 400, 480, 640, 720, 800, 960, 1040, 1200}}), with the latter sort of functioning as a chromatic scale of the former. Smitonic contains a singular perfect fifth, with a major and minor third available on the same note; in general, smitonic contains many intervals of the 5-limit and 7-limit.

The fourth set of scales is the tritone-generated scales: pentic ({{Interval ruler|15|0, 80, 560, 640, 720, 1200}}), peltonic ({{Interval ruler|15|0, 80, 160, 560, 640, 720, 800, 1200}}), balzano ({{Interval ruler|15|0, 80, 160, 240, 560, 640, 720, 800, 880, 1200}}), and semiquinary ({{Interval ruler|15|0, 80, 160, 240, 320, 560, 640, 720, 800, 880, 960, 1200}}).

Surprisingly, the structure of peltonic is exactly analogous to that of 12edo's diatonic! It contains four perfect fifths, which correspond to 12edo's major sixths; 12edo's perfect fifths correspond to large tritones. Its extension, balzano, which I find the most reasonable of the four tritone-generated scales to use, has six fifths, all of which can be used to build some kind of chord.

After this, we have two special families of scales: augmented and blackwood. We will start off with blackwood.

==== Blackwood ====
Blackwood is based on our "fifth-generated diatonic" {{{Interval ruler|15|0, 240, 480, 720, 960, 1200}}}. If we split each step of this scale into a larger and smaller portion, we get the blackwood scale proper: {{{Interval ruler|15|0, 160, 240, 400, 480, 640, 720, 880, 960, 1120, 1200}}}. This scale has only two modes: major and minor, but it is extremely significant if you're used to diatonic harmony, since it has the unique property of always having either a major triad or a minor triad on any given note. In fact, by adding "missing" notes to zarlino to ensure that (which happen to themselves be in the shape of a major chord), we get blackwood. Thus, the blackwood scale can serve as an extremely powerful alternative to the absent MOS diatonic of this edo - it is, in a sense, a "purer" version of said system. (In other edos, this scale's relation to zarlino generalizes to that of [[blackdye]], which loses some of its structural elegance in favor of a more direct diatonic connection and more in-tune intervals.)

==== Augmented ====
Augmented is, conversely, based on major thirds. The basic augmented scale is {{{Interval ruler|15|0, 400, 800, 1200}}}, which is equivalent to 12edo's augmented triad. We can follow the same formula as blackwood and split this scale into large and small steps to get the following possible combinations:

- {{Interval ruler|15|0, 320, 400, 720, 800, 1120, 1200}} - the augmented scale proper, which contains fifths on exactly three of the six notes. 12edo contains a tuning of this scale, but we usually don't talk about it in 12edo.

- {{Interval ruler|15|0, 240, 400, 640, 800, 1040, 1200}} - the wholetone scale, which alternates the two sizes of wholetones and which contains many 7-limit intervals, but no fifths. This is similar to 12edo's wholetone scale, but since there are two different kinds of wholetones in 15edo, it uses both.

- {{Interval ruler|15|0, 160, 320, 400, 560, 720, 800, 960, 1120, 1200}} - the hyrulic scale, which is a way of combining the two prior scales to form a 9-note scale. This can be thought of as having perfect thirds, and major and minor fifths.

- {{Interval ruler|15|0, 240, 320, 400, 640, 720, 800, 1040, 1120, 1200}} - the tcherepnin scale, which is another combination of hard and soft triwood. This can also be thought of as having similar harmony to hyrulic, but with the minor fifth one step flatter.

=== MODMOS structures ===
Let's take the minor scale:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}} - minor

A common way to alter the minor scale is to sharpen the sixth and/or seventh degrees, resulting in the harmonic and melodic minor and variations of Dorian:

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1120, 1200}} - harmonic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1120, 1200}} - dark melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1120, 1200}} - bright melodic minor

{{Interval ruler|15|0, 240, 320, 480, 720, 880, 1040, 1200}}- "diatonyx" dorian

{{Interval ruler|15|0, 240, 320, 480, 720, 960, 1040, 1200}} - didymian dorian

A scale with a single generator (octave-equivalent, of course) is called a "MOS", and all these scales, including normal zarlino, are "MODMOSes" of '''onyx''', formed by sharping or flatting a few notes of it respectively. While technically, because this is an equal tuning, any 7-note scale can be a MODMOS of onyx, I try to reserve it for scales that have harmonic commonalities with diatonic or with onyx.

=== Constructing chords and splitting steps ===
A more interesting way to construct scales is to stack chords rather than single intervals. For example, if we stack 3 major chords, we get:

{{Interval ruler|15|0, 400, 720, 1200}} (major chord on root)

+ {{Interval ruler|15|240, 720, 1120}} (major chord on fifth)

+ {{Interval ruler|15|0, 480, 880, 1200}} (major chord descending from root)

= {{Interval ruler|15|0, 240, 400, 480, 720, 880, 1120, 1200}} - major

Surprise! Here's the zarlino scale again.

Let's try minor chords. This corresponds to flatting the seventh, third, and sixth degrees by one step.

{{Interval ruler|15|0, 240, 320, 480, 720, 800, 1040, 1200}}- minor

In fact, this is actually how the normal major and minor scales are constructed in diatonic, so it's neat to see that it checks out here.

So instead, let's try a different chord (let's say, for some reason, you cared a lot about wolf chords).

{{Interval ruler|15|0, 400, 640, 1200}} (wolf major chord on root)

+ {{Interval ruler|15|80, 640, 1040}} (wolf major chord on wolf fifth)

+ {{Interval ruler|15|0, 560, 960, 1200}} (wolf major chord descending from root)

= {{Interval ruler|15|0, 80, 400, 560, 640, 960, 1040, 1200}} - wolf major

You can call this the "wolf major scale" because of how it's constructed.

Let's try a kind of diminished chord, called a wolf diminished chord, made of two different sizes of minor third. Since the chord is a lot narrower, we'll need 4 chords instead of 3 to build a reasonable scale.

{{Interval ruler|15|0, 240, 560, 1200}} (wolf diminished chord on root)

+ {{Interval ruler|15|560, 800, 1120}} (wolf diminished chord on dim fifth)

+ {{Interval ruler|15|0, 640, 880, 1200}} (wolf diminished chord descending from root)

+ {{Interval ruler|15|80, 320, 640}} (wolf diminished chord descending from aug fourth)

= {{Interval ruler|15|0, 80, 240, 320, 560, 640, 800, 880, 1120, 1200}}- diminished

This scale also contains a "major wolf diminished" chord, which corresponds to the harmonic series sequence 5:6:7, much like a normal major chord corresponds to 4:5:6.

As can be seen, there's a lot of possibilities here.

Another way to build scales is by choosing a few key intervals, and splitting the jumps between them into steps. For example, we might decide that our key notes are {{{Interval ruler|15|0, 400, 480, 720, 1200}}}.

Here, we have two large jumps that can be split into steps, and there's a couple ways to do this, including ones that just result in zarlino and ones that result in scales we haven't seen before.

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1200}}- zaretan

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1200}} - legatus

{{Interval ruler|15|0, 240, 400, 480, 720, 960, 1040, 1200}}- decurion

{{Interval ruler|15|0, 160, 400, 480, 720, 960, 1040, 1200}} - kaiser

If we choose a different set of key intervals, we get a different set of possible scales:

{{Interval ruler|15|0, 240, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 400, 800, 640, 960, 1200}} - anhedonia

{{Interval ruler|15|0, 240, 400, 880, 720, 540, 960, 1200}} - mok

{{Interval ruler|15|0, 560, 400, 960, 1200}} =

{{Interval ruler|15|0, 240, 560, 400, 720, 960, 1200}}- amsel

{{Interval ruler|15|0, 320, 560, 400, 1120, 880, 960, 1200}} - drossel

=== Periodicity blocks ===
In 15edo, there are four primary ways of reaching the semitone on the harmonic table, corresponding to different copies of the semitone repeating periodically near the unison. It can be found as the difference between a stack of two fifths and a minor tone (the syntonic comma), the difference between a major third and a perfect fourth (the limma), the difference between the two tritones (the diaschisma), and the difference between a major and minor third (the classical chroma).

The standard zarlino scale can be defined by having the classical chroma and syntonic comma as its "chromas": if we create a tile that repeats at each multiple of these two chromas, the intervals within that tile are precisely those in the zarlino scale.

However, we can choose a different pair of semitones as our chromas, and get a different scale. Here's the classical chroma x diaschisma scale.

{{Interval ruler|15|0, 240, 320, 400, 560, 640, 800, 960, 1040, 1120, 1200}} - elena

{{Interval ruler|15|0, 240, 320, 400, 560, 720, 800, 960, 1040, 1120, 1200}} - kee'ra*

''*Note: in just intonation, the [[Collection of scales|kee'ra]] scale is a 7-limit "chair" scale, not a 5-limit periodicity block.''

There are two variants as, since the diaschisma can be created by stacking two augmented fourths, the augmented fourth and diminished fifth fall precisely on the edge.

And the classical chroma x limma scale:

{{Interval ruler|15|0, 320, 560, 800, 1040, 1200}}- myn

=== Regular temperaments ===
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 19edo, and Blackwood with 10edo.

== Notation ==
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

{{Cat|Edos}}