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[[File:31edo whr.png|thumb|418x418px|31edo supports [[Carlos Alpha]] and [[Miracle]], alongside supporting Meantone.]]
'''31edo''', or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 39 cents. Aside from [[12edo]], it is a popular tuning of [[Meantone]] and has accurate approximations of harmonics 5 and 7.
'''31edo''', or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 39 cents. Aside from [[12edo]], it is a popular tuning of [[Meantone]] and has accurate approximations of harmonics 5 and 7.


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== Notation ==
== 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.
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
|1,006.45
|nearminor seventh
|minor seventh
|G
|9/5, 16/9
|-
|27
|1,045.16
|neutral seventh
|neutral seventh
|Gt
|11/6, 20/11
|-
|28
|1,083.87
|nearmajor seventh
|major seventh
|G#
|15/8
|-
|29
|1,122.58
|supermajor seventh
|semiaugmented seventh
|Ab
|48/25
|-
|30
|1,161.29
|suboctave
|semidiminished octave
|Ad
|49/25, 125/64, 96/49
|-
|31
|1,200.00
|octave
|octave
|A
|2/1
|}
{{Cat|Edos}}{{Navbox EDO}}

Latest revision as of 02:23, 14 February 2026

31edo supports Carlos Alpha and Miracle, alongside supporting Meantone.

31edo, or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 39 cents. Aside from 12edo, it is a popular tuning of Meantone and has accurate approximations of harmonics 5 and 7.

Theory

Edostep interpretations

31edo's edostep has the following interpretations in the 2...13 subgroup:

  • 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)
  • 65/64 (the difference between 16/13 and 5/4)

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").


Approximation of prime harmonics in 31edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) 0.0 -5.2 +0.8 -1.1 -9.4 +11.1 +11.2 +12.2 -8.9 +15.6 +16.3
Relative (%) 0.0 -13.4 +2.0 -2.8 -24.2 +28.6 +28.9 +31.4 -23.0 +40.3 +42.0
Steps

(reduced)

31

(0)

49

(18)

72

(10)

87

(25)

107

(14)

115

(22)

127

(3)

132

(8)

140

(16)

151

(27)

154

(30)

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

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.

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 Meantone/monocot.

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).

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.

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 1,006.45 nearminor seventh minor seventh G 9/5, 16/9
27 1,045.16 neutral seventh neutral seventh Gt 11/6, 20/11
28 1,083.87 nearmajor seventh major seventh G# 15/8
29 1,122.58 supermajor seventh semiaugmented seventh Ab 48/25
30 1,161.29 suboctave semidiminished octave Ad 49/25, 125/64, 96/49
31 1,200.00 octave octave A 2/1


View • Talk • EditETs
EDOs
Macrotonal 5 • 7 • 8 • 9 • 10 • 11
12-23 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 21 • 22 • 23
24-35 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35
36-47 36 • 37 • 39 • 41 • 43 • 44 • 45 • 46 • 47
48-59 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58
60-71 60 • 63 • 64 • 65 • 67 • 70
72-83 72 • 77 • 81
84-95 84 • 87 • 89 • 90 • 93 • 94
Large EDOs 99 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 224 • 239 • 270 • 306 • 311 • 612 • 665
Nonoctave equal temperaments
Tritave 4 • 9 • 13 • 17 • 26 • 39
Fifth 8 • 9
Other


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