Xenharmonic Reference - User contributions [en]

31edo

2026-03-28T22:17:35Z

Lériendil: /* 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
|'''310'''
|348
|'''387'''
|426
|-
!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.

=== 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, 31edo has a large number of full-period MOS scales that exist in the edo. 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
|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
|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
|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
|}

== 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

2026-03-28T22:14:58Z

Lériendil: /* 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
|'''310'''
|348
|'''387'''
|426
|-
!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.

=== 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, 31edo has a large number of full-period MOS scales that exist in the edo. 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
|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
|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
|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
|}

== 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

2026-03-28T20:16:34Z

Lériendil: /* 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 is best understood as a 2.3.5.7.11.23 system, although it has a sharp but functional prime 13. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval.

It has a rather functional diatonic scale, with the whole tone split into 2 and 3, with 2 steps making a chromatic semitone and 3 steps making a diatonic semitone.

{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality ([[ADIN]])
|Subminor
|'''Nearminor'''
|Neutral
|'''Nearmajor'''
|Supermajor
|-
!Cents
|271
|'''310'''
|348
|'''387'''
|426
|-
!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.

=== 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, 31edo has a large number of full-period MOS scales that exist in the edo. 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
|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
|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
|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
|}

== 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

2026-03-28T19:30:05Z

Lériendil: /* 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 is best understood as a 2.3.5.7.11.23 system, although it has a sharp but functional prime 13. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval. It has a rather functional diatonic scale, with the whole tone split into 2 and 3, with 2 steps making a chromatic semitone and 3 steps making a diatonic semitone.
{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality ([[ADIN]])
|Subminor
|'''Nearminor'''
|Neutral
|'''Nearmajor'''
|Supermajor
|-
!Cents
|271
|'''310'''
|348
|'''387'''
|426
|-
!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.

=== 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, 31edo has a large number of full-period MOS scales that exist in the edo. 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
|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
|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
|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
|}

== 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}}

Adaptive diatonic interval names

2026-03-28T15:47:11Z

Lériendil: /* Alternative system */

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Extensions ==

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

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

31edo

2026-03-28T14:40:15Z

Lériendil: /* Edostep interpretations */

[[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 is best understood as a 2.3.5.7.11.23 system, although it has a sharp but functional prime 13. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval. It has a rather functional diatonic scale, with the whole tone split into 2 and 3, with 2 steps making a chromatic semitone (or "chromatone") and 3 steps making a diatonic semitone (or "diatone").

{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality
|Subminor
|'''Pentaminor'''
|Neutral
|'''Pentamajor'''
|Supermajor
|-
!Cents
|271
|'''310'''
|348
|'''387'''
|426
|-
!Just interpretation
|7/6
|'''6/5'''
|11/9
|'''5/4'''
|9/7
|-
!Steps
|7
|'''8'''
|9
|'''10'''
|11
|}
Diatonic thirds are bolded.

=== 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, 31edo has a large number of full-period MOS scales that exist in the edo. 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
|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
|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
|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
|}

== 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

2026-03-28T14:40:06Z

Lériendil: /* Theory */

[[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]])
* 100/99 (the difference between 11/10 and 10/9)

==== JI approximation ====
31edo is best understood as a 2.3.5.7.11.23 system, although it has a sharp but functional prime 13. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval. It has a rather functional diatonic scale, with the whole tone split into 2 and 3, with 2 steps making a chromatic semitone (or "chromatone") and 3 steps making a diatonic semitone (or "diatone").

{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality
|Subminor
|'''Pentaminor'''
|Neutral
|'''Pentamajor'''
|Supermajor
|-
!Cents
|271
|'''310'''
|348
|'''387'''
|426
|-
!Just interpretation
|7/6
|'''6/5'''
|11/9
|'''5/4'''
|9/7
|-
!Steps
|7
|'''8'''
|9
|'''10'''
|11
|}
Diatonic thirds are bolded.

=== 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, 31edo has a large number of full-period MOS scales that exist in the edo. 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
|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
|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
|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
|}

== 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

2026-03-28T10:25:24Z

Lériendil:

[[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:

* 49/48 (the difference between 7/6 and 8/7)
* 50/49 (the difference between 7/5 and 10/7)
* 64/63 (the difference between 8/7 and 9/8)
* 36/35 (the difference between 7/6 and 6/5)
* 54/55 (the difference between 6/5 and 11/9)
* 45/44 (the difference between 5/4 and 11/9)
* 128/125 (the difference between 5/4 and 32/25)

==== JI approximation ====
31edo is best understood as a 2.3.5.7.11.23 system, although it has a sharp but functional prime 13. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval. It has a rather functional diatonic scale, with the whole tone split into 2 and 3, with 2 steps making a chromatic semitone (or "chromatone") and 3 steps making a diatonic semitone (or "diatone").

{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality
|Subminor
|'''Pentaminor'''
|Neutral
|'''Pentamajor'''
|Supermajor
|-
!Cents
|271
|'''310'''
|348
|'''387'''
|426
|-
!Just interpretation
|7/6
|'''6/5'''
|11/9
|'''5/4'''
|9/7
|-
!Steps
|7
|'''8'''
|9
|'''10'''
|11
|}
Diatonic thirds are bolded.

=== 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, 31edo has a large number of full-period MOS scales that exist in the edo. 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
|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
|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
|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
|}

== 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

2026-03-28T10:24:47Z

Lériendil:

[[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]], formed by stacking [[8/7]], and emphasizing 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:

* 49/48 (the difference between 7/6 and 8/7)
* 50/49 (the difference between 7/5 and 10/7)
* 64/63 (the difference between 8/7 and 9/8)
* 36/35 (the difference between 7/6 and 6/5)
* 54/55 (the difference between 6/5 and 11/9)
* 45/44 (the difference between 5/4 and 11/9)
* 128/125 (the difference between 5/4 and 32/25)

==== JI approximation ====
31edo is best understood as a 2.3.5.7.11.23 system, although it has a sharp but functional prime 13. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval. It has a rather functional diatonic scale, with the whole tone split into 2 and 3, with 2 steps making a chromatic semitone (or "chromatone") and 3 steps making a diatonic semitone (or "diatone").

{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality
|Subminor
|'''Pentaminor'''
|Neutral
|'''Pentamajor'''
|Supermajor
|-
!Cents
|271
|'''310'''
|348
|'''387'''
|426
|-
!Just interpretation
|7/6
|'''6/5'''
|11/9
|'''5/4'''
|9/7
|-
!Steps
|7
|'''8'''
|9
|'''10'''
|11
|}
Diatonic thirds are bolded.

=== 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, 31edo has a large number of full-period MOS scales that exist in the edo. 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
|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
|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
|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
|}

== 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

2026-03-28T10:21:08Z

Lériendil: added proper intro section

[[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. Dividing the fifth in two gives us a neutral-third temperament, in this case [[Mohajira]], while dividing it in three gives us [[Slendric]]; 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: Mohajira emphasizing the 11th harmonic and Slendric emphasizing the 7th, with Miracle and Valentine being all-rounders. [[Orwell]], generated by 31edo's subminor third, gives yet another way to encompass the 11-limit. 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:

* 49/48 (the difference between 7/6 and 8/7)
* 50/49 (the difference between 7/5 and 10/7)
* 64/63 (the difference between 8/7 and 9/8)
* 36/35 (the difference between 7/6 and 6/5)
* 54/55 (the difference between 6/5 and 11/9)
* 45/44 (the difference between 5/4 and 11/9)
* 128/125 (the difference between 5/4 and 32/25)

==== JI approximation ====
31edo is best understood as a 2.3.5.7.11.23 system, although it has a sharp but functional prime 13. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval. It has a rather functional diatonic scale, with the whole tone split into 2 and 3, with 2 steps making a chromatic semitone (or "chromatone") and 3 steps making a diatonic semitone (or "diatone").

{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality
|Subminor
|'''Pentaminor'''
|Neutral
|'''Pentamajor'''
|Supermajor
|-
!Cents
|271
|'''310'''
|348
|'''387'''
|426
|-
!Just interpretation
|7/6
|'''6/5'''
|11/9
|'''5/4'''
|9/7
|-
!Steps
|7
|'''8'''
|9
|'''10'''
|11
|}
Diatonic thirds are bolded.

=== 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, 31edo has a large number of full-period MOS scales that exist in the edo. 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
|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
|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
|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
|}

== 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}}

40edo

2026-03-12T13:18:26Z

Lériendil: added sw links

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three [[20/19]]'s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several 10:13:16:19 tetrads available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament. While the whole tone in Deeptone approximates 10/9 very well, it cannot be interpreted that way outside of the dual-prime interpretation.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell/Guanyintet ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes. This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

An alternative to Orwell[13] worth mentioning is "[https://scaleworkshop.plainsound.org/scale/pwxnuq0Gb Orwell[14]]", constructed by splitting, rather than the large step of Orwell[9], the small step into a 1\40 chroma and a remainder. This can be considered an [[aberrismic]] superset of Orwell[9], and due to the 5:4 hardness of Orwell[9], consists of step sizes 5\, 3\, and 1\40 that all differ by the same amount, 2\40. This is a subset of Orwell[22] that manages to reach the higher harmonies found in the Guanyintet chain.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

==== 13edo-derived muddles ====
A notable scale of 40edo is generated by 3\40, stacking thrice to the Orwell generator of 7/6 and nine times to 8/5. The principal generated [[MOS]] is of 13 steps - twelve being length 3\40 and one being length 4\40, and serving as a [[well-temperament]] of [[13edo]] - and contains multiple instances of the 10:13:16:19 tetrad. As this MOS scale is quite nearly an EDO, one is incentivized to take further subsets of it that have useful melodic and harmonic properties - that is, [[MOS muddle]]s.

As taking every 3 generators essentially gives Orwell, the most notable generator chains of 13edo reflected in 40edo in this manner include the stacks of 4\13 and 5\13, representing [[3L 4s]]/[[3L 7s]], and [[oneirotonic]] respectively. [https://scaleworkshop.plainsound.org/scale/JxxQLHZ5Q The former sequence] is an easy approach to incorporating 10:13:16:19 into a musically coherent scale, as it occurs even in the 7-note muddle if the 4\40 step is placed correctly. The latter provides a simulation of the oneirotonic scale in the largest EDO to formally lack one.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-12T13:10:16Z

Lériendil: redundant with "structural chains"

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three [[20/19]]'s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several 10:13:16:19 tetrads available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament. While the whole tone in Deeptone approximates 10/9 very well, it cannot be interpreted that way outside of the dual-prime interpretation.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell/Guanyintet ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

==== 13edo-derived muddles ====
A notable scale of 40edo is generated by 3\40, stacking thrice to the Orwell generator of 7/6 and nine times to 8/5. The principal generated [[MOS]] is of 13 steps - twelve being length 3\40 and one being length 4\40, and serving as a [[well-temperament]] of [[13edo]] - and contains multiple instances of the 10:13:16:19 tetrad. As this MOS scale is quite nearly an EDO, one is incentivized to take further subsets of it that have useful melodic and harmonic properties - that is, [[MOS muddle]]s.

As taking every 3 generators essentially gives Orwell, the most notable generator chains of 13edo reflected in 40edo in this manner include the stacks of 4\13 and 5\13, representing [[3L 4s]]/[[3L 7s]], and [[oneirotonic]] respectively. The former sequence is an easy approach to incorporating 10:13:16:19 into a musically coherent scale, as it occurs even in the 7-note muddle if the 4\40 step is placed correctly. The latter provides a simulation of the oneirotonic scale in the largest EDO to formally lack one.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-12T13:09:44Z

Lériendil: /* 13edo-derived muddles */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three [[20/19]]'s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several 10:13:16:19 tetrads available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament. While the whole tone in Deeptone approximates 10/9 very well, it cannot be interpreted that way outside of the dual-prime interpretation.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell/Guanyintet ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

==== 13edo-derived muddles ====
A notable scale of 40edo is generated by 3\40, stacking thrice to the Orwell generator of 7/6 and nine times to 8/5. The generator itself approximates 20/19, while four and five generators, respectively, approximate 13/10 and 16/13. The principal generated [[MOS]] is of 13 steps - twelve being length 3\40 and one being length 4\40, and serving as a [[well-temperament]] of [[13edo]] - and contains multiple instances of the 10:13:16:19 tetrad. As this MOS scale is quite nearly an EDO, one is incentivized to take further subsets of it that have useful melodic and harmonic properties - that is, [[MOS muddle]]s.

As taking every 3 generators essentially gives Orwell, the most notable generator chains of 13edo reflected in 40edo in this manner include the stacks of 4\13 and 5\13, representing [[3L 4s]]/[[3L 7s]], and [[oneirotonic]] respectively. The former sequence is an easy approach to incorporating 10:13:16:19 into a musically coherent scale, as it occurs even in the 7-note muddle if the 4\40 step is placed correctly. The latter provides a simulation of the oneirotonic scale in the largest EDO to formally lack one.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-12T13:09:30Z

Lériendil: /* Tempered commas */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three [[20/19]]'s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several 10:13:16:19 tetrads available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament. While the whole tone in Deeptone approximates 10/9 very well, it cannot be interpreted that way outside of the dual-prime interpretation.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell/Guanyintet ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

==== 13edo-derived muddles ====
A notable scale of 40edo is generated by 3\40, stacking thrice to the Orwell generator of 7/6 and nine times to 8/5. The generator itself approximates [[20/19]], while four and five generators, respectively, approximate 13/10 and 16/13. The principal generated [[MOS]] is of 13 steps - twelve being length 3\40 and one being length 4\40, and serving as a [[well-temperament]] of [[13edo]] - and contains multiple instances of the 10:13:16:19 tetrad. As this MOS scale is quite nearly an EDO, one is incentivized to take further subsets of it that have useful melodic and harmonic properties - that is, [[MOS muddle]]s.

As taking every 3 generators essentially gives Orwell, the most notable generator chains of 13edo reflected in 40edo in this manner include the stacks of 4\13 and 5\13, representing [[3L 4s]]/[[3L 7s]], and [[oneirotonic]] respectively. The former sequence is an easy approach to incorporating 10:13:16:19 into a musically coherent scale, as it occurs even in the 7-note muddle if the 4\40 step is placed correctly. The latter provides a simulation of the oneirotonic scale in the largest EDO to formally lack one.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-12T13:09:00Z

Lériendil: added 3\40 genseq

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several 10:13:16:19 tetrads available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament. While the whole tone in Deeptone approximates 10/9 very well, it cannot be interpreted that way outside of the dual-prime interpretation.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell/Guanyintet ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

==== 13edo-derived muddles ====
A notable scale of 40edo is generated by 3\40, stacking thrice to the Orwell generator of 7/6 and nine times to 8/5. The generator itself approximates [[20/19]], while four and five generators, respectively, approximate 13/10 and 16/13. The principal generated [[MOS]] is of 13 steps - twelve being length 3\40 and one being length 4\40, and serving as a [[well-temperament]] of [[13edo]] - and contains multiple instances of the 10:13:16:19 tetrad. As this MOS scale is quite nearly an EDO, one is incentivized to take further subsets of it that have useful melodic and harmonic properties - that is, [[MOS muddle]]s.

As taking every 3 generators essentially gives Orwell, the most notable generator chains of 13edo reflected in 40edo in this manner include the stacks of 4\13 and 5\13, representing [[3L 4s]]/[[3L 7s]], and [[oneirotonic]] respectively. The former sequence is an easy approach to incorporating 10:13:16:19 into a musically coherent scale, as it occurs even in the 7-note muddle if the 4\40 step is placed correctly. The latter provides a simulation of the oneirotonic scale in the largest EDO to formally lack one.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

MediaWiki:Timeless.css

2026-03-10T20:34:48Z

Lériendil: Undoing the change for the moment; the infobox problem on light mode is worse than overriding manual color changes

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Slendric

2026-03-10T20:31:55Z

Lériendil:

{{Infobox regtemp
| Title = Slendric
| Subgroups = 2.3.7
| Comma basis = [[1029/1024]]
| Edo join 1 = 5 | Edo join 2 = 21
| Mapping = 1; 3 -1
| Generators = 8/7 | Generators tuning = 233.7 | Optimization method = CWE
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], …
| Odd limit 1 = 7 | Mistuning 1 = 2.11 | Complexity 1 = 11
| Odd limit 2 = 2.3.7 27 | Mistuning 2 = 2.81 | Complexity 2 = 21
}}
'''Slendric''' (also known as "Wonder" or "Gamelic") is the basic harmonic interpretation as a [[regular temperament]] for the structure where the perfect fifth (~[[3/2]]) is split into three equal parts; each of these is taken to represent the interval [[8/7]]. Since the 7th [[harmonic]] is less than 3 [[cent]]s from just when 3/2 is pure, Slendric constitutes an exceptionally good [[rank-2 temperament|rank-2]] traversal of the [[2.3.7 subgroup|2.3.7]] tuning space for its simplicity. Its corresponding [[comma]] is the difference between 3/2 and (8/7)3, which is 1029/1024.

Melodically, the Slendric generator stack forms a 5-note scale (1L 4s) that is nearly [[equipentatonic]]. [[MOS]]es further down the hierarchy (6, 11, 16, ... notes) can be thought of as the notes of a basic pentatonic form, inflected by multiples of a characteristic small interval known as the ''quark'' (representing a third of a [[diatonic semitone]], and the commas [[49/48]] and [[64/63]] tempered together). As a result, these MOS scales tend to be extremely [[hard]].

Slendric can exhibit a wide range of tunings, with fifths between those of [[26edo]] (692c) and [[56edo]] (707c), or generators roughly between 231 and 236c, while maintaining the recognizability of the 2.3.7 structure. Notable [[EDO]] tunings are in between these, and include EDOs that end in "1" or "6", i.e. [[31edo]], [[36edo]], [[41edo]], and [[46edo]]. Slendric is also supported by edos with 5edo's 3/2.

== Structural theory ==
=== General theory ===
==== Interval categories ====
It is possible to define the intervals of Slendric in terms of diatonic categories, for at three steps is the perfect fifth, and at every three steps further are all of the standard fifth-generated intervals. For the remaining steps, a single pair of inflections suffices: "up"/"down", which can be abbreviated with the prefixes S and s, respectively (standing in for "super" and "sub", which can be used synonymously). An "up" is rigorously defined to be an inflection by the "quark" of 49/48~64/63. The slendric generator is then the upmajor second, and therefore the 2-generator interval is a downfourth (as a major second and a perfect fourth together reach a perfect fifth) as well as a double-upmajor third. Between a major third and perfect fourth is a minor second, which is therefore equivalent to three repetitions of "up"; because of this equivalence, it is never necessary to attach more than one "up"/"down" to a diatonic interval.

Note that "up" intervals and "down" intervals can be represented as fractions with a single factor of 7 in the denominator and numerator (compactly, "/7" or "ru", and "7/" or "zo" intervals), respectively, with uninflected diatonic intervals representing the [[3-limit]]. Considering extensions to prime 5, Rodan maps 7/5 onto the [[chain of fifths]] so that "up" and "down" also comprise the /5 and 5/ classes of intervals, while Mothra maps 5 directly onto the chain of fifths. Each of these provides a very intuitive way to notate the full [[7-limit]].

==== The pentatonic framework ====
The intervals of Slendric can be organized according to how many steps of [[5edo]], or equivalently the 5-note MOS, they correspond to, since the MOS scales of Slendric up to at least 26 notes have 5 large steps and many small steps, each the size of a quark. The "major" interval of a class here is the one just larger than the corresponding 5edo interval, and the "minor" interval is just smaller. Below are the intervals of the symmetric mode of Slendric[21] (5L 16s). The generator tuning here is 3/10-comma, where the quark is exactly sqrt([[28/27]]), or about 31.5 cents.

{| class="wikitable center-all left-1" style="color: #000000;"
|-
! Steps of 5edo
!0
!1
!2
!3
!4
!5
|- style="background-color: #DFDFDF;"
! "Augmented" interval
| 63.0
| 296.7
| 530.4
| 764.1
| 997.8
|
|- style="background-color: #F7F7F7;"
! JI intervals represented
| 28/27
| 32/27
| 49/36
| 14/9
| 16/9
|
|- style="background-color: #DFDFDF;"
! "Major" interval
| 31.5
| 265.2
| 498.9
| 732.6
| 966.3
| ''1200.0''
|- style="background-color: #F7F7F7;"
! JI intervals represented
| 49/48, 64/63
| 7/6
| 4/3
| 32/21, 49/32
| 7/4
| ''2/1''
|- style="background-color: #DFDFDF;"
! "Minor" interval
| ''0.0''
| 233.7
| 467.4
| 701.1
| 934.8
| 1168.5
|- style="background-color: #F7F7F7;"
! JI intervals represented
| ''1/1''
| 8/7
| 21/16, 64/49
| 3/2
| 12/7
| 63/32, 96/49
|- style="background-color: #DFDFDF;"
! "Diminished" interval
|
| 202.2
| 435.9
| 669.6
| 903.3
| 1137.0
|- style="background-color: #F7F7F7;"
! JI intervals represented
|
| 9/8
| 9/7
| 72/49
| 27/16
| 27/14
|}

=== Notable features and related structures ===
A distinctive feature of Slendric tuning systems is the subfourth of two generators, which represents [[21/16]]. Additionally, it serves as (8/7)2 = 64/49, and thus is tempered a few cents flat of 21/16 in most tunings. Another interpretation then is [[17/13]], tempering out 273/272 and 833/832, into which 1029/1024 factors. (31edo's tuning comes particularly close to 17/13.)

As a result of the ease of finding 55/32 and 17/13 along the Slendric chain, any extension to the full 7-limit can also find prime 11, and any extension to 2.3.7.13 can also find prime 17. This applies both to strong and weak extensions.
==== A-Team ====
Taking every other step of Slendric results in a subtemperament generated by this subfourth, which is known as '''A-Team''' (5 & 18; the name is from "18"). It is one of the main regular temperaments representing the [[oneirotonic]] (5L 3s) scale - specifically the hard tunings thereof such as in [[18edo|18]], [[23edo|23]], and [[31edo]]. The core [[subgroup]] interpretation of A-Team is 2.9.21.55: note that two A-Team generators, representing [[12/7]], come close to [[55/32]] and therefore [[385/384]] and [[441/440]], which again multiply to 1029/1024, can be tempered out. Different A-Team tunings can pick up other harmonic approximations; an interesting one is the 13:17:19 chord found in Mothra (and especially 31edo)'s version of A-Team.

A-Team may be considered a [[straddle primes|straddle-3]] temperament; the >3 is the sharp 32/21 generator and the <3 is reached by +4 21/16 generators. Indeed, stacking >3 and <3 reaches 9 at +3 generators.

Two subranges of A-Team are
* 2.9.5.21[13 & 18], a weak restriction of Mothra, equating +6 oneirotonic generators with 5/4, thus tempering out 81/80
* 2.9.15.21[18 & 23] (sometimes known as '''B-Team'''), a weak restriction of Rodan, equating the diminished 3-oneirostep (-7 oneirotonic generators) with 6/5

==== Relationship with acoustic phi ====
{{adv|The A-Team generator acquires the representations 21/16, 17/13, [[55/42]], and [[72/55]]. But if we look one octave higher, a pattern becomes clear: [[21/8]], [[34/13]], [[55/21]], and [[144/55]] are all ratios of two-apart Fibonacci numbers, which therefore closely approximate the square of [[acoustic phi]], entailing that acoustic phi squared over 2 is close to two Slendric generators. A single generator, therefore, is approximable by acoustic phi divided by [[sqrt(2)]]. This can also be explained by 18 being the 6th Lucas number, and therefore a close approximation to φ6; approximating 181/6 by φ gives us φ/√2 as an approximation of (3/2)1/3. This interval's precise value is about 233.09{{c}}, and using it as a generator produces a form of Slendric too sharp to be Mothra but flat of 36edo, with a fifth about 2.7 cents flat.}}

=== Interval chain ===
In the following tables, odd harmonics and subharmonics 1–27 are labeled in '''bold'''. Cent values reflect 3/10-comma tuning.

<div><div style="display: inline-grid; margin-right: 25px;">

{| class="wikitable sortable center-1 center-2 right-3"
|-
! #
! class="unsortable" | Extended diatonic category
! Cents
! class="unsortable" | Approximate 2.3.7 ratios
|-
| 0
| P1
| 0
| '''1/1'''
|-
| 1
| SM2
| 234
| '''8/7'''
|-
| 2
| s4
| 467
| '''21/16''', 64/49
|-
| 3
| P5
| 701
| '''3/2'''
|-
| 4
| SM6
| 935
| 12/7
|-
| 5
| s8
| 1169
| 63/32, 96/49
|-
| 6
| M2
| 202
| '''9/8'''
|-
| 7
| SM3
| 436
| 9/7
|-
| 8
| s5
| 670
| 72/49
|-
| 9
| M6
| 903
| '''27/16'''
|-
| 10
| SM7
| 1137
| 27/14
|-
| 11
| sM2
| 171
| 54/49
|}
</div>
<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable sortable center-1 center-2 right-3"
|-
! #
! class="unsortable" | Extended diatonic category
! Cents
! class="unsortable" | Approximate 2.3.7 ratios
|-
| 0
| P1
| 0
| '''1/1'''
|-
| −1
| sm7
| 966
| '''7/4'''
|-
| −2
| S5
| 733
| '''32/21''', 49/32
|-
| −3
| P4
| 499
| '''4/3'''
|-
| −4
| sm3
| 265
| 7/6
|-
| −5
| S1
| 31
| 49/48, 64/63
|-
| −6
| m7
| 998
| '''16/9'''
|-
| −7
| sm6
| 764
| 14/9
|-
| −8
| S4
| 530
| 49/36
|-
| −9
| m3
| 297
| '''32/27'''
|-
| −10
| sm2
| 63
| 28/27
|-
| −11
| Sm7
| 1029
| 49/27
|}
</div>

== Tunings and extensions ==
=== Tuning considerations ===
The error induced by the comma 1029/1024, about 8.4{{c}}, has to be distributed between three factors of 7 and one factor of 3, and ideally both 3 and 7 should be flattened; we can define tunings of Slendric by the fraction of this comma by which 8/7 is sharpened. As representations of 2.3.7 intervals generally stack more factors of 3 than factors of 7, it can be argued 3 should be flattened less than 7. This occurs between 1/3-comma tuning (234.0{{c}}, just flat of 41edo) which sets 3/2, and thus the entire Pythagorean chain, just while 8/7 is sharpened by 2.8{{c}}; and 1/4-comma tuning (233.3{{c}}, very close to 36edo) which sets them equally flat, so that [[7/6]] is just. A notable EDO tuning in this range is [[77edo]].

But, especially if [[6:7:8]] is considered the fundamental 2.3.7 harmony, it is reasonable to want a tuning where the error of [[4/3]] is split between that of 7/6 and 8/7. Furthermore, sharpward error is often considered more acceptable than flatward error on the interval 7/6, and these flatter tunings of Slendric are those which happen to tune 7/6 sharp. 1/5-comma tuning (232.9{{c}}, very near [[67edo]]) sets 7/6 and 8/7 equally sharp, by about 1.7{{c}} each.

Based on the above, 36edo can be considered a practically optimal tuning, as it is an EDO of reasonable size in the best range for pure 2.3.7 subgroup accuracy; however, it is essentially [[straddle primes|straddle]]-5 and straddle-11, being between two full [[11-limit]] interpretations (the 36p and 36ce [[val]]s). Thus, other tunings of Slendric should be sought to improve the accuracy of [[5-limit]] and 11-limit harmony.

Additional particularities of Slendric to consider include the tuning of the subfourth, and the size of the quark. The subfourth varies between nearly [[13/10]] in the flattest tunings, which has a potentially tertian function (e.g. in 10:13:15 [[triad]]s), and near-just 21/16 in the sharpest tunings, which much more closely resembles a fourth; of the intervals of Slendric, this is the one with the least clear independent role and the most variability in function between the different tunings. As for the quark, its size can vary between that of a comma and that of a quartertone. Tunings where the fifth is flattened significantly (specifiable as Mothra in the full 7-limit) have a more melodically salient quark that serves as an [[aberrisma]], and bring [[7/4|the 7th harmonic]] closer to purity.

=== Extensions ===
As Slendric is a structure present in very many EDOs of note and an obvious simplification of the 2.3.7 subgroup aside from that, it behooves us to consider how this structure interacts with other harmonies within the [[17-limit]]. Fortunately, there are a variety of choices for how each of the primes 5, 11, 13, and 17 fits into the Slendric framework.

==== Prime 5 ====
There are two most important strong extensions to reach prime 5 and complete the 7-limit, these being ''Mothra'' and ''Rodan''. Other mappings of 5 can be employed, such as the [[Schismic]] one (known as ''Guiron''), but these are significantly more complex and give prime 5 less structural presence.

Mothra uses a [[meantone]] fifth in order to find [[5/4]] at the diatonic major third (12 generators up) and temper out [[81/80]]. The exaggerated quark now represents [[36/35]] in addition to 49/48 and 64/63. The most important Mothra tunings are 31edo, at the optimum for this temperament with a close-to-just 5/4, and 26edo, which approximates the tuning formed by stacking a pure 8/7. 36edo using the [[12edo]] major third of 400{{c}} as 5/4 also qualifies as Mothra.

Rodan, meanwhile, slightly sharpens the fifth and can be constructed by equating 81/80 to the quark. This thereby tempers out the [[aberschisma]] (5120/5103), and furthermore implies the Sensamagic ([[245/243]]) equivalence, that [[9/7]] forms half of [[5/3]]. From this, it can be seen that 5/4 is found at a perfect fifth (3 generators) above twice 9/7 (7 generators each), or 17 generators in all: this is the downmajor third in [[#Interval categories|the system described earlier]]. 41edo and 46edo bound the main Rodan tuning range, but their sum, [[87edo]], is essentially optimal with a nearly just 5/4. 36edo using the flat major third of 367{{c}} as 5/4 also qualifies as Rodan.

As regards weak extensions, notable ones include [[Miracle]], which splits 8/7 in two, and [[Valentine]], which splits it in three (therefore dividing the perfect fifth into 6 and 9 parts, respectively). They also include [[Superkleismic]] (15 & 26), which splits 7/4 into three intervals of [[6/5]]; this is supported by 26, 41, and 56edo.

Miracle's generator (known as the "secor") represents [[16/15]] and [[15/14]] simultaneously (tempering out [[225/224]], the marvel comma), so that [[8/5]] is placed at 7 secors. As a consequence of splitting 8/7 in half, Miracle also includes an exact [[neutral third]], interpretable in the 7-limit as [[49/40]]. Miracle is 10 & 21, and 31, 41, and [[72edo]] support it.

Valentine places 6/5 and 5/4 at 4 and 5 steps respectively; the generator thus represents (5/4)/(6/5) = [[25/24]] and (6/5)/(8/7) = [[21/20]], and their ratio ([[126/125]], the starling comma) is tempered out. Valentine is 15 & 16, and 31, 46, and 77edo support it.

==== Prime 11 ====
As mentioned before, extensions to 11 can be created off of these by tempering out 385/384 and 441/440. This works almost perfectly in the 41 & 46 Rodan range, and the diatonic major third is identified with [[14/11]]. This also applies to the 26 & 31 Mothra range, yet the case with Mothra is slightly more complicated, as the interval formed from (sharpened) 7/6 stacked twice can reasonably represent either [[11/8]] or [[15/11]], depending on the tuning (note that in 31edo, it represents both).

The former is supported by 26 & 31, and the latter by 31 & 36; the resulting extensions are called "undecimal Mothra" and "Mosura" respectively. Undecimal Mothra equates 14/11 to 9/7 (tempering out [[99/98]]), and Mosura equates 14/11 to [[32/25]] (tempering out [[176/175]]). Of the two, the former is taken to be canonical primarily as 11/8 itself is reached by far fewer generators (-8, compared to 23).

Miracle, Valentine, and Superkleismic all receive extensions to 11 in this manner as well. In Miracle's case, the neutral third is mapped to [[11/9]]~[[27/22]], while in Valentine's, the [[neutral second]] formed by two steps represents [[12/11]]~[[11/10]]. Superkleismic, in fact, tempers out [[100/99]], whereby [[16/11]] is reached at only two of its 6/5 generators, which produces the notable 2.7.11 subgroup structure known as [[Orgone]].

==== Primes 13 and 17 ====
While 36edo's representation of the 2.3.7 subgroup fails to provide comparably accurate harmonies of 5 and 11, it does somewhat better with the next higher primes: 13, 17, and 19 (though the latter two descend from 12edo). Looking at 36edo's mapping of 13, we see that it divides 7/6 into halves that can each be taken as [[14/13]]~[[13/12]], and further that two quarks represent 28/27 and [[27/26]] simultaneously.

The former leads us to a weak extension, known as ''Baladic'', that tempers out [[169/168]] and splits the octave in two; equating 17/13 to the downfourth, we see [[9/8]] is also split into [[18/17]]~[[17/16]], and therefore that [[17/12]] is a semioctave.

The latter leads us to a strong extension, called ''Euslendric'' (36 & 77), that reaches 13/8 after 19 generators, as the up-augmented fifth, and 17/16 after 21 generators, as the augmented unison. Euslendric is notable as its harmonies can extend to even higher limits, reaching [[19/16]] as the minor third (-9 generators), [[23/16]] as the up-diminished fifth (-23 generators), and [[29/16]] as the upminor seventh (-11 generators), all within the optimal tuning band for 2.3.7 accuracy.

Revisiting the 11-limit extensions mentioned above, Rodan naturally obtains [[13/11]] as the minor third to find 13 at 22 generators down. Meanwhile, Mothra's 9/8 is flat enough that it is very close to 143/128 = (11/8)/([[16/13]]), so [[144/143]] can be tempered out as a way to extend each 11-limit extension of Mothra further to the [[13-limit]]. All of these take on the obvious mapping to reach the full 17-limit, though in the case of Rodan, 17 receives greater damage than any lower prime in the most accurate Rodan tunings (such as 87edo).

=== Tuning spectrum ===
[[File:Slendric Tuning Chart.png|thumb|alt=Slendric Tuning Chart.png|A chart of the tuning spectrum of Slendric, showing the offsets of odd harmonics 3, 7, 9, and 21, as a function of the generator. All EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some important EDOs supporting the temperament are labeled. Comma fractions with corresponding unchanged intervals are also labeled.]]

{| class="wikitable center-all left-4 left-5"
|-
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]*
! Generator (¢)
! Mapping of 5
! Comments
|-
| '''[[11edo|2\11]]'''
|
| '''218.182'''
|
| '''Lower bound of {1, 3, 7, 9} diamond monotone'''
|-
| [[16edo|3\16]]
|
| 225.000
| ↓ +7 gens "Gorgo" {36/35}
|
|-
| [[37edo|7\37]]
|
| 227.027
|
| 37b val
|-
| [[21edo|4\21]]
|
| 228.571
| ↑ Gorgo ↓ -14 gens "Archaeotherium" {405/392}
|
|-
| [[47edo|9\47]]
|
| 229.787
|
|
|-
| [[26edo|5\26]]
|
| 230.769
| ↑ Archaeotherium ↓ +12 gens "[[Mothra#Tuning spectrum|Mothra]]" {81/80}
|
|-
|
| [[8/7]]
| 231.174
|
| Untempered tuning
|-
| [[57edo|11\57]]
|
| 231.579
|
|
|-
|
| [[17/13]]
| 232.214
|
| As s4, approx. 1/8-comma
|-
| [[31edo|6\31]]
|
| 232.258
|
|
|-
| [[67edo|13\67]]
|
| 232.836
|
|
|-
|
| [[96/49]]
| 232.861
|
| 1/5-comma
|-
|
| φ/√2
| 233.090
|
| As generator
|-
|
| [[12/7]]
| 233.282
|
| 1/4-comma; (2.3.7) 7-odd-limit minimax tuning
|-
| [[36edo|7\36]]
|
| 233.333
| ↑ Mothra ↓ -24 gens "Guiron" {10976/10935}
|
|-
|
| [[9/7]]
| 233.583
|
| 2/7-comma; (2.3.7) 9-odd-limit minimax tuning
|-
| [[113edo|22\113]]
|
| 233.628
|
| 113c val (guiron)
|-
|
| [[27/14]]
| 233.704
|
| 3/10-comma; 2.3.7 [[CEE]] tuning
|-
| [[77edo|15\77]]
|
| 233.766
|
|
|-
| [[118edo|23\118]]
|
| 233.898
|
|
|-
|
| [[3/2]]
| 233.985
|
| 1/3-comma; (2.3.7) 21- and 27-odd-limit minimax tuning
|-
| [[41edo|8\41]]
|
| 234.146
| ↑ Guiron ↓ +17 gens "[[Rodan#Tuning spectrum|Rodan]]" {245/243}
|
|-
|
| [[55/32]]
| 234.408
|
| As SM6, approx. 3/8-comma
|-
| [[87edo|17\87]]
|
| 234.483
|
|
|-
|
| [[63/32]]
| 234.547
|
| 2/5-comma
|-
| [[46edo|9\46]]
|
| 234.783
| ↑ Rodan
|
|-
| [[97edo|19\97]]
|
| 235.052
|
|
|-
| [[51edo|10\51]]
|
| 235.294
|
|
|-
|
| [[21/16]]
| 235.390
|
| 1/2-comma
|-
| [[56edo|11\56]]
|
| 235.714
|
|
|-
| [[61edo|12\61]]
|
| 236.066
|
|
|-
| [[66edo|13\66]]
|
| 236.364
|
|
|-
| [[71edo|14\71]]
|
| 236.620
|
|
|-
| '''[[5edo|1\5]]'''
|
| '''240.000'''
|
| '''Upper bound of {1, 3, 7, 9} diamond monotone'''
|}
<nowiki>*</nowiki> Besides the octave

{{Navbox regtemp}}
{{cat|Temperaments}}

40edo

2026-03-10T13:37:51Z

Lériendil: /* Notable structural chains */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several 10:13:16:19 tetrads available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament. While the whole tone in Deeptone approximates 10/9 very well, it cannot be interpreted that way outside of the dual-prime interpretation.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell/Guanyintet ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-10T13:35:57Z

Lériendil: /* Notable structural chains */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several 10:13:16:19 tetrads available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell/Guanyintet ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-10T13:35:21Z

Lériendil: /* Orwell */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several tetrads 10:13:16:19 available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell/Guanyintet ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-10T13:35:02Z

Lériendil: /* Notable structural chains */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270{{c}}) comprise ~8/5 (810{{c}}), and furthermore that two intervals of 7/6 comprise ~15/11 (540{{c}}). This is [[Guanyintet]] temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several tetrads 10:13:16:19 available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690{{c}}) is of note. Interpreting the fifth as 3/2, the major third (formally ~[[81/64]]) is mapped to 16/13, and as the apotome is the narrow 30{{c}} difference between it and the minor third (~[[39/32]]), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of [[Deeptone]] temperament.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

Adaptive diatonic interval names

2026-03-10T03:19:47Z

Lériendil: /* Alternative system */

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Extensions ==

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

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

Adaptive diatonic interval names

2026-03-10T03:19:40Z

Lériendil: /* Alternative system */

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Extensions ==

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

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

Guanyintet

2026-03-09T17:21:08Z

Lériendil: Redirected page to Orwell#Guanyintet

#redirect [[Orwell #Guanyintet]]

40edo

2026-03-09T17:14:52Z

Lériendil: /* 120edo */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-08T15:59:44Z

Lériendil: /* Tempering properties */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

=== Notable structural chains ===
{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of [[12edo]]. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-08T15:52:37Z

Lériendil:

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of [[12edo]]. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-08T15:52:31Z

Lériendil:

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 cents exactly, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of [[12edo]]. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-08T15:52:00Z

Lériendil: /* Tempered commas */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

The dual-{3 7 11 17} interpretation additionally tempers out the following:
* [[136/135]] (diatismic), S16*S17, equating 9/8 with 17/15
* [[289/288]] (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
* [[361/360]] (dudon), S19, and [[400/399]] (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
* [[390625/388962]] (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave.

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of [[12edo]]. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-08T15:46:48Z

Lériendil: /* Intervals and notation */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8''', 17/15
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33, 39/34
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19, 68/39
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9, 30/17
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

In its patent 13-limit, 40edo tempers out:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave

{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of [[12edo]]. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-08T15:42:47Z

Lériendil: /* Tempered commas */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

In its patent 13-limit, 40edo tempers out:
* [[66/65]], S11*S12 (winmeanmic), equating 6/5 and [[13/11]]
* [[99/98]] (mothwellsmic), equating 9/7 and 14/11
* [[105/104]], S14*S15 (animist), equating 8/7 and 15/13
* [[121/120]], S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to [[11/10]]
* [[225/224]], S15 (marvel), splitting 8/7 into [[15/14]]~[[16/15]] and equating a stack of two 5/4s to [[14/9]]
* [[648/625]] (diminished), setting 6/5 to the quarter-octave
* [[1053/1024]] (superflat), making 16/13 the diatonic major third
* [[2187/2080]], placing 5/4 an apotome above 16/13 (making it the augmented third)
* [[16807/16384]] (cloudy), setting 8/7 to a fifth of the octave

{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of [[12edo]]. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

200edo

2026-03-08T15:31:00Z

Lériendil: Redirected page to 40edo#200edo

#redirect [[40edo #200edo]]

120edo

2026-03-08T15:30:53Z

Lériendil: Redirected page to 40edo#120edo

#redirect [[40edo #120edo]]

80edo

2026-03-08T15:30:43Z

Lériendil: Redirected page to 40edo#80edo

#redirect [[40edo #80edo]]

40edo

2026-03-08T15:30:16Z

Lériendil: /* Multiples */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

== Multiples ==
As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

=== 80edo ===
Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64{{c}} flat), 9/7 (0.087{{c}} flat), [[17/16]] (0.045{{c}} sharp), and most incredibly, 11/10 (0.004{{c}} flat), and as a tuning for temperaments such as [[Diaschismic]] and [[Echidna]].
{{Harmonics in ED|80|31|0}}

=== 120edo ===
120edo splits the octave into three, and includes the familiar 700{{c}} fifth of [[12edo]]. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of [[Myna]].
{{Harmonics in ED|120|31|0}}

=== 200edo ===
200edo's most notable feature is its highly accurate [[3/2]], being the smallest EDO with a better approximation thereof than [[53edo]]. It, somewhat conveniently, splits this fifth into nine, allowing it to tune [[Carlos Alpha]]. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from [[50edo]] rather than 40.
{{Harmonics in ED|200|31|0}}

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

40edo

2026-03-08T15:13:34Z

Lériendil:

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsmic), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8
* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

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

40edo

2026-03-08T15:10:12Z

Lériendil: /* Tempering properties */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
=== Tempered commas ===
Important [[comma]]s tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:
* [[176/175]], S8/S10 (valinorsma), equating a stack of two 5/4s to [[11/7]]
* [[456/455]] (abnobisma), equating (5/4)*(7/6) to [[19/13]]
* [[540/539]], S12/S14 (swetisma), equating two 7/6s to [[15/11]]
* [[1728/1715]], S6/S7 (orwellisma), equating three 7/6s to 8/5
* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
* [[48013/48000]], S19/S20, splitting 7/6 into three 20/19s.

{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating [[14/11]] and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and [[19/16]] as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

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

User:Vector/Diesis

2026-03-08T06:21:24Z

Lériendil: Lériendil moved page User:Vector/diesis to User:Vector/Diesis without leaving a redirect

{| class="wikitable"
|+
!interval
!factor
!diesis
|-
|5
|
|81/80
|-
|7
|
|64/63
|-
|11
|
|33/32
|-
|11
|
|513216/524288
|-
|13
|
|1053/1024
|-
|17
|
|4131/4096
|-
|19
|
|513/512
|-
|23
|
|736/729
|-
|23
|
|16767/16384
|-
|29
|
|261/256
|-
|31
|
|32/31
|-
|31
|
|248/243
|-
|37
|
|37/36
|-
|37
|
|1024/999
|-
|41
|
|82/81
|-
|43
|
|129/128
|-
|47
|
|48/47
|-
|53
|
|54/53
|-
|59
|
|243/236
|-
|61
|
|244/243
|-
|67
|
|16384/16281
|-
|71
|
|72/71
|-
|73
|
|73/72
|-
|79
|
|81/79
|-
|83
|
|83/81
|-
|83
|
|256/249
|-
|89
|
|729/712
|-
|89
|
|65536/64881
|-
|97
|
|97/96
|-
|25
|5*5
|2048/2025
|-
|35
|7*5
|36/35
|-
|55
|11*5
|55/54
|-
|65
|13*5
|65/64
|-
|85
|17*5
|256/255
|-
|95
|19*5
|96/95
|-
|115
|23*5
|1035/1024
|-
|49
|7*7
|49/48
|-
|49
|7*7
|4096/3969
|-
|77
|11*7
|2079/2048
|-
|91
|13*7
|729/728
|-
|91
|13*7
|66339/65536
|-
|119
|17*7
|243/238
|-
|133
|19*7
|32768/32319
|-
|161
|23*7
|162/161
|-
|121
|11*11
|243/242
|-
|143
|13*11
|144/143
|-
|187
|17*11
|192/187
|-
|187
|17*11
|748/729
|-
|169
|13*13
|512/507
|-
|145
|29*5
|145/144
|-
|155
|31*5
|4185/4096
|-
|185
|37*5
|134865/131072
|-
|185
|37*5
|740/729
|-
|7/5
|
|5120/5103
|-
|7/5
|
|3645/3584
|-
|11/5
|
|45/44
|-
|13/5
|
|40/39
|-
|13/5
|
|416/405
|-
|17/5
|
|136/135
|-
|19/5
|
|1216/1215
|-
|23/5
|
|46/45
|-
|23/5
|
|640/621
|-
|11/7
|
|896/891
|-
|13/7
|
|1701/1664
|-
|17/7
|
|459/448
|-
|19/7
|
|57/56
|-
|23/7
|
|189/184
|-
|13/11
|
|352/351
|-
|17/11
|
|34/33
|-
|17/11
|
|1408/1377
|-
|29/5
|
|21141/20480
|-
|29/5
|
|3712/3645
|-
|31/5
|
|31/30
|-
|31/5
|
|2560/2511
|-
|37/5
|
|1215/1184
|-
|125
|5*5*5
|128/125
|-
|125
|5*5*5
|250/243
|-
|175
|5*5*7
|525/512
|-
|175
|5*5*7
|131072/127575
|-
|275
|5*5*11
|66825/65536
|-
|325
|5*5*13
|325/324
|-
|245
|5*7*7
|245/243
|-
|385
|5*7*11
|385/384
|-
|343
|7*7*7
|1029/1024
|-
|55/7
|11*5/7
|56/55
|-
|49/5
|7*7/5
|405/392
|-
|77/5
|11*7/5
|1232/1215
|-
|25/7
|5*5/7
|225/224
|-
|25/11
|5*5/11
|100/99
|-
|25/13
|5*5/13
|3200/3159
|-
|35/11
|5*7/11
|2835/2816
|-
|625
|5*5*5*5
|16875/16384
|}

40edo

2026-03-08T01:12:06Z

Lériendil: /* Intervals and notation */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating [[14/11]] and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and [[19/16]] as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

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

40edo

2026-03-08T00:25:18Z

Lériendil: /* Deeptone diatonic */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating [[14/11]] and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and [[19/16]] as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone ====
40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

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

40edo

2026-03-07T23:59:46Z

Lériendil: /* Compositional theory */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Tempering properties ==
{{WIP}}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating [[14/11]] and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and [[19/16]] as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone diatonic ====
{{WIP}}

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

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

40edo

2026-03-07T23:58:29Z

Lériendil: /* Orwell */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating [[14/11]] and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and [[19/16]] as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone diatonic ====
{{WIP}}

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
The fundamental scale of 40edo's Orwell temperament is the 9-note scale, [[4L 5s]], with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690{{c}} fifths occur in the enneatonic, far more common are 660{{c}} 22/15~35/24~19/13 subfifths. {{adv|It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24{{c}}).}}

The 9-note scale is followed by a 13-note chromatic ([[9L 4s]]) with steps of 4\40 and 1\40. These cluster severely around [[9edo]], however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an [[aberrismic]] scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the [[2L 5s]] scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of [[pelog]] tunings.

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

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

40edo

2026-03-07T22:21:50Z

Lériendil: /* Scales */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating [[14/11]] and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and [[19/16]] as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone diatonic ====
{{WIP}}

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
{{WIP}}

==== Diminished and Blackwood ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

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

40edo

2026-03-07T22:21:23Z

Lériendil: /* Scales */

'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave [[2/1]].

40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80|syntonic comma]] is mapped ''negatively'' in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's [[11-limit]] is a tuning for [[Orwell|undecimal Orwell]], and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note [[MOS]], [[4L 5s]] - most notably 40edo's approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp.

== General theory ==
=== JI approximation ===
While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).
{{Harmonics in ED|40|prime}}

=== Edostep interpretations ===
In the 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:
* 56/55 (the difference between 5/4 and [[14/11]])
* 57/56 (the difference between 7/6 and [[19/16]])
* 65/64 (the difference between 16/13 and 5/4)
* 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:
* 50/49 (the difference between [[49/40]] and 5/4, or 7/6 and [[25/21]])
* 55/54 (the difference between [[12/11]] and [[10/9]])
* 81/80 (the difference between 10/9 and [[9/8]])
* 105/104 (the difference between [[13/10]] and [[21/16]]);
alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:
* 27/26 (the difference between 10/9 and [[15/13]])
* 33/32 (the difference between [[4/3]] and [[11/8]])
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and [[9/7]])
* 45/44 (the difference between [[11/9]] and 5/4)
* 49/48 (the difference between 8/7 and 7/6)
* 80/81 (the ''negative'' difference between 9/8 and 10/9);
alongside the first set of representations.

=== Intervals and notation ===
As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.
{| class="wikitable"
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="3" |JI approximations
! colspan="2" |Notation
|-
! rowspan="1" |2.5.7/3.11/3.13.19.23 subgroup
! rowspan="1" |Dual-{3 7 11 17}
! rowspan="1" |Patent 13-limit val
! rowspan="1" |Native-fifths
! rowspan="1" |Orwell
|-
|0
|0
|1/1
|
|
|D
|J
|-
|1
|30
|[56/55], [57/56], '''65/64'''
|50/49, 51/50, 52/51
|'''''33/32''''', ''36/35'', 45/44, 49/48
|D#
|J#
|-
|2
|60
|26/25
|'''33/32''', 35/34
|''21/20''
|Dx
|Jx, Kbb
|-
|3
|90
|[20/19]
|52/49, 19/18, 21/20
|22/21, ''25/24''
|D#x, Ebbb
|Kb
|-
|4
|120
|[15/14]
|49/46
|14/13, 16/15
|Ebb
|K
|-
|5
|150
|[12/11], 25/23
|23/21
|''11/10'', 13/12
|Eb
|K#
|-
|6
|180
|39/35
|10/9, [51/46], 21/19
|'''''9/8'''''
|E
|Kx
|-
|7
|210
|26/23, [44/39]
|'''9/8'''
|''10/9''
|E#
|Lbb
|-
|8
|240
|[23/20], 55/48
|38/33
|15/13, 8/7
|Ex
|Lb
|-
|9
|270
|7/6
|
|
|Fbb
|L
|-
|10
|300
|'''19/16'''
|[25/21]
|''6/5'', 13/11
|Fb
|L#
|-
|11
|330
|23/19
|17/14, 40/33
|
|F
|Lx, Mbb
|-
|12
|360
|[16/13]
|26/21, 49/40
|11/9
|F#
|Mb
|-
|13
|390
|44/35, '''5/4'''
|64/51
|
|Fx
|M
|-
|14
|420
|32/25, 14/11
|23/18, [51/40], 33/26
|''9/7''
|F#x, Gbbb
|M#
|-
|15
|450
|13/10
|64/49, 22/17, 49/38
|
|Gbb
|Mx
|-
|16
|480
|25/19
|[33/25], '''21/16'''
|
|Gb
|Nbb
|-
|17
|510
|66/49
|51/38
|4/3
|G
|Nb
|-
|18
|540
|48/35, 26/19, 15/11
|
|'''11/8'''
|G#
|N
|-
|19
|570
|39/28, [32/23]
|46/33, [25/18], 18/13
|7/5
|Gx
|N#
|-
|20
|600
|55/39, 78/55
|17/12, 24/17
|
|G#x, Abbb
|Nx, Obb
|-
|21
|630
|['''23/16'''], 56/39
|13/9, [36/25], 33/23
|10/7
|Abb
|Ob
|-
|22
|660
|22/15, 19/13, 35/24
|
|16/11
|Ab
|O
|-
|23
|690
|49/33
|76/51
|'''3/2'''
|A
|O#
|-
|24
|720
|38/25
|32/21, [50/33]
|
|A#
|Ox
|-
|25
|750
|20/13
|76/49, 17/11, '''49/32'''
|
|Ax
|Pbb
|-
|26
|780
|11/7, '''25/16'''
|52/33, 80/51, 36/23
|''14/9''
|A#x, Bbbb
|Pb
|-
|27
|810
|8/5, 35/22
|'''51/32'''
|
|Bbb
|P
|-
|28
|840
|['''13/8''']
|80/49, 21/13
|18/11
|Bb
|P#
|-
|29
|870
|38/23
|33/20, 28/17
|
|B
|Px, Qbb
|-
|30
|900
|32/19
|[42/25]
|22/13, ''5/3''
|B#
|Qb
|-
|31
|930
|12/7
|
|
|Bx
|Q
|-
|32
|960
|96/55, [40/23]
|33/19
|'''7/4''', 26/15
|Cbb
|Q#
|-
|33
|990
|[39/22], 23/13
|16/9
|''9/5''
|Cb
|Qx
|-
|34
|1020
|70/39
|38/21, [92/51], 9/5
|''16/9''
|C
|Rbb
|-
|35
|1050
|46/25, [11/6]
|42/23
|24/13, ''20/11''
|C#
|Rb
|-
|36
|1080
|[28/15]
|92/49
|'''15/8''', 13/7
|Cx
|R
|-
|37
|1110
|[19/10]
|40/21, 36/19, 49/26
|
|C#x, Dbbb
|R#
|-
|38
|1140
|25/13
|68/35, 64/33
|
|Dbb
|Rx, Jbb
|-
|39
|1170
|128/65, [112/57], [55/28]
|51/26, 100/51, 49/25
|
|Db
|Jb
|-
|40
|1200
|2/1
|
|
|D
|J
|}

== Compositional theory ==
=== Tertian structure ===
Six intervals in 40edo can be considered functional "[[third]]s" with the 690{{c}} diatonic fifth taken as the bounding interval; a seventh (450{{c}}) can be included with the acknowledgement of the 720{{c}} blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating [[14/11]] and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and [[19/16]] as a pair of fifth complements, if the diatonic fifth is used.

{| class="wikitable"
|+Thirds in 40edo
!Quality ([[ADIN]])
|Subminor
|Nearminor
|'''Supraminor'''
|'''Submajor'''
|Nearmajor
|Supermajor
|Ultramajor
|-
!Cents
|270
|300
|'''330'''
|'''360'''
|390
|420
|450
|-
!Just interpretation
|7/6 (+3.1{{c}})
|19/16 (+2.5{{c}})
|'''23/19 (-0.8{{c}})'''
|'''16/13 (+0.5{{c}})'''
|5/4 (+3.7{{c}})
|14/11 (+2.5{{c}})
|13/10 (-4.2{{c}})
|-
!Steps
|9
|10
|'''11'''
|'''12'''
|13
|14
|15
|}
Diatonic thirds are bolded.

=== Chords ===
In addition to triads bounded by a perfect fifth, in 40edo one finds that 810{{c}} (8/5) and 660{{c}} (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic [[7-limit]] tetrad [[4:5:6:7]]. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660{{c}} interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330{{c}} supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting [[otonal]] representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a [[chthonic harmony|chthonic]] triad, of which Orwell[9] provides two additional variants stacked within the [[perfect fourth]] (17\40).

Lastly, 40edo contains a nearly-[[isoharmonic]] diminished triad, similarly to [[22edo]], at [0 11 20]\40, approximating 24:29:34.

=== Scales ===
As 40edo is composite, it incorporates the scales of all of its subset EDOs, including [[8edo]], [[10edo]], and [[20edo]]. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

==== Deeptone diatonic ====
{{WIP}}

==== Omnidiatonic/Diasem ====
Consider the [[Zarlino]] diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as [[diasem]].

Additionally, these scales are [[chiral]], so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

==== Orwell ====
{{WIP}}

==== Multi-period scales ====
Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90{{c}} semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called [[Diminished]]. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90{{c}} semitone above a quarter-octave, taken together, these imply that the flat fifth (690{{c}}) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480{{c}}, identified with [[21/16]]) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called [[Blackwood]]. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90{{c}} semitone ''below'' the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

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

40edo

2026-03-07T21:49:18Z

Lériendil: /* Blackwood */

40edo

2026-03-07T21:49:02Z

Lériendil: /* Scales */

40edo

2026-03-07T21:48:42Z

Lériendil: /* Scales */

40edo

2026-03-07T21:43:52Z

Lériendil: /* Scales */

40edo

2026-03-07T20:26:44Z

Lériendil: /* Edostep interpretations */

40edo

2026-03-06T21:02:01Z

Lériendil: /* Edostep interpretations */