31edo

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31edo
Prime factorization 31 (prime)
Step size 38.7097¢
Fifth 18\31 (696.8¢)
Major 2nd 5\31 (193.5¢)
Semitones (A1:m2) 2:3 (77.4¢ : 116.1¢)
Consistency limit 11
English Wikipedia has an article on:

31 equal divisions of the octave (31edo), or 31(-tone) equal temperament, tricesimoprimal meantone temperament (31tet, 31et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 31 equally large steps. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.7 cents. The term tricesimoprimal was first used by Adriaan Fokker.

Basic theory

Approximation of prime intervals in 31 EDO
Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 -5.2 +0.8 -1.1 -9.4 +11.1 +11.2 +12.2 -8.9
relative (%) +0 -13 +2 -3 -24 +29 +29 +31 -23
Steps (reduced) 31 (0) 49 (18) 72 (10) 87 (25) 107 (14) 115 (22) 127 (3) 132 (8) 140 (16)

31edo's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of quarter-comma meantone. 31's approximation of 7/4, a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the 11-limit (and specifically the 11-odd-limit), although the fact that it equates 14/11 with 9/7 and 11/8 with 15/11 could potentially be considered too much tuning damage. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).

Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31edo is relatively quite accurate and is the 6th zeta integral edo, the 7th zeta gap EDO, a zeta peak EDO and a zeta peak integer EDO, meaning it is a strict zeta EDO. Another way in which 31edo is especially accurate is that it represents a record in Pepper ambiguity in the 7-, 9- and 11-odd-limit, which it is consistent through.

One step of 31edo, measuring about 38.7¢, is called a diesis because it stands in for several intervals called "dieses" (such as 128/125 and 648/625) which are tempered out in 12edo. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in SpiralProgressions. Zhea Erose's 31edo music uses the interval frequently.

31edo is close to a circle made by stacking 31 pure 17/13 subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents (relative error 7.1%).

31edo is the 11th prime EDO, following 29edo and coming before 37edo.

Intervals

Main article: 31edo/Individual degrees
See also: Table of 31edo intervals
Degree Cents Approximate Ratios Ups and Downs Notation Extended pythagorean notation
0 0.00 1/1 P1 perfect unison D P1 perfect unison D
1 38.71 45/44, 49/48, 46/45, 128/125, 36/35 ^1, d2 up-unison, dim 2nd ^D, Ebb d2 dim 2nd Ebb
2 77.42 25/24, 21/20, 22/21, 23/22 A1, vm2 aug 1sn, downminor 2nd D#, vEb A1 aug 1sn D#
3 116.13 15/14, 16/15 m2 minor 2nd Eb m2 minor 2nd Eb
4 154.84 12/11, 11/10, 35/32 ~2 mid 2nd vE AA1, dd3 double-aug 1sn, double-dim 3rd Dx, Fbb
5 193.55 9/8, 10/9, 19/17, 28/25 M2 major 2nd E M2 major 2nd E
6 232.26 8/7, 144/125 ^M2 upmajor 2nd ^E d3 dim 3rd Fb
7 270.97 7/6, 75/64 vm3 downminor 3rd vF A2 aug 2nd E#
8 309.68 6/5, 25/21 m3 minor 3rd F m3 minor 3rd F
9 348.39 11/9, 27/22, 16/13, 60/49, 49/40 ~3 mid 3rd ^F AA2, dd4 double-aug 2nd, double-dim 4th Ex, Gbb
10 387.10 5/4 M3 major 3rd F# M3 major 3rd F#
11 425.81 9/7, 14/11, 23/18, 32/25 ^M3 upmajor 3rd ^F# d4 dim 4th Gb
12 464.52 21/16, 13/10, 17/13, 125/96 v4 down-4th vG A3 aug 3rd Fx
13 503.23 4/3 P4 perfect 4th G P4 perfect 4th G
14 541.94 11/8, 15/11, 26/19 ^4, ~4 up-4th, mid 4th ^G AA3, dd5 double-aug 3rd, double-dim 5th Fx#, Abb
15 580.65 7/5, 45/32, 25/18 A4, vd5 aug 4th, downdim 5th G#, vAb A4 aug 4th G#
16 619.35 10/7, 64/45, 36/25 ^A4, d5 upaug 4th, dim 5th ^G#, Ab d5 dim 5th Ab
17 658.06 16/11, 22/15, 19/13 v5, ~5 down-5th, mid 5th vA AA4, dd6 double-aug 4th, double-dim 6th Gx, Bbbb
18 696.77 3/2 P5 perfect 5th A P5 perfect 5th A
19 735.48 32/21, 20/13, 26/17, 192/125 ^5 up-5th ^A d6 dim 6th Bbb
20 774.19 14/9, 11/7, 25/16 vm6 downminor 6th vBb A5 aug 5th A#
21 812.90 8/5 m6 minor 6th Bb m6 minor 6th Bb
22 851.61 18/11, 44/27, 13/8, 49/30, 80/49 ~6 mid 6th vB AA5, dd7 double-aug 5th, double-dim 7th Ax, Cbb
23 890.32 5/3, 42/25 M6 major 6th B M6 major 6th B
24 929.03 12/7, 128/75 ^M6 upmajor 6th ^B d7 dim 7th Cb
25 967.74 7/4, 125/72 vm7 downminor 7th vC A6 aug 6th B#
26 1006.45 16/9, 9/5, 34/19, 25/14 m7 minor 7th C m7 minor 7th C
27 1045.16 11/6, 20/11, 64/35 ~7 mid 7th ^C AA6, dd8 double-aug 6th, double-dim 8ve Bx, Dbb
28 1083.87 28/15, 15/8 M7 major 7th C# M7 major 7th C#
29 1122.58 48/25, 40/21, 21/11, 44/23 ^M7 upmajor 7th ^C# d8 dim 8ve Db
30 1161.29 88/45, 96/49, 45/23, 125/64, 35/18 v8 down-8ve vD A7 aug 7th Cx
31 1200.00 2/1 P8 perfect 8ve D P8 perfect 8ve D

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality color name monzo format examples
downminor zo [a b 0 1 7/6, 7/4
minor fourthward wa [a b where b < -1 32/27, 16/9
gu [a b -1 6/5, 9/5
mid ilo [a b 0 0 1 11/9, 11/6
lu [a b 0 0 -1 12/11, 18/11
major yo [a b 1 5/4, 5/3
fifthward wa [a b where b > 1 9/8, 27/16
upmajor ru [a b 0 -1 9/7, 12/7

All 31edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:

color of the 3rd JI chord EDO-steps notes of C chord written name spoken name
zo 6:7:9 0-7-18 C vEb G Cvm C downminor
gu 10:12:15 0-8-18 C Eb G Cm C minor
ilo 18:22:27 0-9-18 C vE G C~ C mid
yo 4:5:6 0-10-18 C E G C C major or C
ru 14:18:21 0-11-18 C ^E G C^ C upmajor or C up

For a more complete list of chords, see 31edo Chord Names and Ups and Downs Notation #Chords and Chord Progressions.

Notations

Neutral circle-of-fifths notation

Neutral circle-of-fifths notation is much like ups and downs notation. Between C and D (do and re) for example, we have the following notes:

Degree Letter Name English full name
0 C do C
1 C+ do+ C half-sharp
2 C# do# C sharp
3 Db re b D flat
4 Dd re d D half-flat
5 D re D

Circle-of-fifths notation

Circle-of-fifths notation uses double sharps and double flats:

Degree Letter Name English full name
0 C do C
1 Dbb re bb D double flat
2 C# do # C sharp
3 Db re b D flat
4 Cx do x C double sharp
5 D re D

While using double sharp and double flat may seem confusing because it alternates between C and D, it provides a way of writing chords that is consistent with traditional notation. For example, the subminor7 chord 12:14:18:21 is written like so:

  • C / D# / G / A#
  • C# / Dx / G# / Ax
  • Db / E / Ab / B
  • D / E# / A / B#

In 12edo, we have the enharmonic equivalences C# = Db and E# = F. But in 31edo we have:

  • Cx = Dd
  • C+ = Dbb
  • E# = Fd
  • E+ = Fb
  • Ex = F+
  • Ed = Fbb

Sagittal notation

The Revo flavor of Sagittal notation from the appendix to The Sagittal Songbook by Jacob A. Barton:

31edo Sagittal.png

JI approximation

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 31edo. Prime harmonics are in bold; inconsistent intervals are in italic.

15-odd-limit intervals by direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢) Error (rel, %)
5/4, 8/5 0.783 2.0
11/9, 18/11 0.979 2.5
7/4, 8/7 1.084 2.8
7/5, 10/7 1.867 4.8
15/14, 28/15 3.314 8.6
7/6, 12/7 4.097 10.6
11/6, 12/11 4.202 10.9
15/8, 16/15 4.398 11.4
15/11, 22/15 4.985 12.9
3/2, 4/3 5.181 13.4
5/3, 6/5 5.964 15.4
11/7, 14/11 8.298 21.4
9/7, 14/9 9.278 24.0
11/8, 16/11 9.382 24.2
11/10, 20/11 10.166 26.3
13/10, 20/13 10.302 26.6
9/8, 16/9 10.362 26.8
13/8, 16/13 11.085 28.6
9/5, 10/9 11.145 28.8
13/7, 14/13 12.169 31.4
15/13, 26/15 15.483 40.0
13/12, 24/13 16.266 42.0
13/9, 18/13 17.263 44.6
13/11, 22/13 18.242 47.1
15-odd-limit intervals by patent val mapping
Interval, complement Error (abs, ¢) Error (rel, %)
5/4, 8/5 0.783 2.0
11/9, 18/11 0.979 2.5
7/4, 8/7 1.084 2.8
7/5, 10/7 1.867 4.8
15/14, 28/15 3.314 8.6
7/6, 12/7 4.097 10.6
11/6, 12/11 4.202 10.9
15/8, 16/15 4.398 11.4
15/11, 22/15 4.985 12.9
3/2, 4/3 5.181 13.4
5/3, 6/5 5.964 15.4
11/7, 14/11 8.298 21.4
9/7, 14/9 9.278 24.0
11/8, 16/11 9.382 24.2
11/10, 20/11 10.166 26.3
13/10, 20/13 10.302 26.6
9/8, 16/9 10.362 26.8
13/8, 16/13 11.085 28.6
9/5, 10/9 11.145 28.8
13/7, 14/13 12.169 31.4
15/13, 26/15 15.483 40.0
13/12, 24/13 16.266 42.0
13/11, 22/13 20.468 52.9
13/9, 18/13 21.447 55.4

Selected 19-limit intervals

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Relationship to 12-edo

Whereas 12-edo has a circle of twelve 5ths, 31-edo has a spiral of twelve 5ths (since 18\31 is on the 7\12 kite in the scale tree). This spiral of 5th shows 31-edo in a 12-edo-friendly format. Excellent for introducing 31-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.

31-edo spiral.png

31edo can be notated with a seperate semi/sesqui sharp/flat chain (like 17edo), with its own enharmonic circle of fifths.

31edo CoF semi and sesqui.png

Scales

MOS scales

Main article: 31edo MOS scales

The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other MOSes and MOS chains[clarification needed] are also useful:

  • 9\31 neutral third generator generates ultrasoft mosh and superhard dicotonic MOSes.
  • 11\31 generator generates a parahard sensoid scale with resolution from neutral thirds, sixths, and sevenths to perfect fourths, fifths, and octaves, and a semihard 3L 8s scale with a jagged-but-chromatic feel.
  • 12\31 generator generates a semihard oneirotonic scale, similar to the 5L 3s scale in 13edo but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth close to 19/13.
  • A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo.
  • If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The ultrasoft 9-tone orwelloid (4L 5s) MOS could be treated as a 9-tone well temperament.
  • It has close approximations to 6edf (→ miracle) and 9edf (→ Carlos Alpha), fifth-equivalent equal divisions that hit many good JI approximations.

See #Rank-2 temperaments for a table of MOSes and their temperament interpretations.

Harmonic scales

31edo approximates Mode 8 of the harmonic series okay, but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated okay, but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the 13-limit.

The steps are: 5 5 4 4 4 3 3 3

Overtones in "Mode 8": 8 9 10 11 12 13 14 15 16
…as JI Ratio from 1/1: 1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8 2/1
…in cents: 0 203.9 386.3 551.3 702.0 840.5 968.8 1088.3 1200.0
Nearest degree of 31edo: 0 5 10 14 18 22 25 28 31
…in cents: 0 193.5 387.1 541.9 696.8 851.6 967.7 1083.9 1200.0

In mode 16, the most closely-matched harmonics are the composite ones, 21 and 25. Of the other harmonics:

  • 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.
  • 19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent (see mercy temperament).
  • 23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent.
  • 27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.
  • 29 and 31 are both very sharp, and intervals involving them are unlikely to play any major role.
Odd overtones in "Mode 16": 17 19 21 23 25 27 29 31
…as JI Ratio from 1/1: 17/16 19/16 21/16 23/16 25/16 27/16 29/16 31/16
…in cents: 105.0 297.5 470.8 628.3 772.6 905.9 1029.6 1145.0
Nearest degree of 31edo: 3 8 12 16 20 23 27 30
…in cents: 116.1 309.7 464.5 619.4 774.2 890.3 1045.1 1161.3

Various subsets

A large open list of subsets from 31edo that people have named:

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢)
Tuning error
Absolute (¢) Relative (%)
2.3 [-49 31 [31 49]] +1.63 1.64 4.22
2.3.5 81/80, 393216/390625 [31 49 72]] +0.98 1.63 4.20
2.3.5.7 81/80, 126/125, 1029/1024 [31 49 72 87]] +0.83 1.43 3.70
2.3.5.7.11 81/80, 99/98, 121/120, 126/125 [31 49 72 87 107]] +1.21 1.49 3.84

31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next equal temperaments doing better in those subgroups are 72, 72, 41, and 46, respectively.

31edo excels in the 2.5.7 subgroup (the JI chord 4:5:7 is represented highly consistently: to distance 10.36). In 2.5.7 it tempers out the didacus comma 3136/3125 and the quince comma 823543/819200, thus also tempering out the very small rainy comma, the simplest 2.5.7 comma tempered out by the 7-limit microtemperament 171edo. In the 11-limit, 31edo can be defined as the unique temperament that tempers out 81/80, 99/98, 121/120 and 126/125, and it supports orwell, mohajira, and the relatively high-accuracy temperament miracle. In the 13-limit 31edo doesn't do as well, but is the optimal patent val for the rank five temperament tempering out the 13-limit comma 66/65, which equates 6/5 and 13/11. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit. In the 17-limit it tempers out 120/119, equating the otonal tetrad of 4:5:6:7 and the inversion of the 10:12:15:17 minor tetrad.

Commas

31edo tempers out the following commas. This assumes the val 31 49 72 87 107 115], comma values rounded to 5 significant digits.

Prime
limit
Ratio[1] Monzo Cents Color name Name
5 (16 digits) [-25 7 6 31.567 Lala-tribiyo Ampersand
5 81/80 [-4 4 -1 21.506 Gu Syntonic comma
5 (12 digits) [17 1 -8 11.445 Saquadbigu Würschmidt comma
5 (14 digits) [-21 3 7 10.061 Lasepyo Semicomma
5 (24 digits) [38 -2 -15 1.3843 Sasa-quintrigu Hemithirds comma
7 (18 digits) [-10 7 8 -7 22.413 Lasepru-aquadbiyo Blackjackisma
7 64827/64000 [-9 3 -3 4 22.227 Laquadzo-atrigu Squalentine
7 2430/2401 [1 5 1 -4 20.785 Quadru-ayo Nuwell
7 50421/50000 [-4 1 -5 5 14.516 Quinzogu Trimyna
7 126/125 [1 2 -3 1 13.795 Zotrigu Starling comma
7 1728/1715 [6 3 -1 -3 13.074 Trizo-agu Orwellisma
7 1029/1024 [-10 1 0 3 8.4327 Latrizo Gamelisma
7 225/224 [-5 2 2 -1 7.7115 Ruyoyo Marvel comma
7 16875/16807 [0 3 4 -5 6.9903 Quinru-aquadyo Mirkwai
7 3136/3125 [6 0 -5 2 6.0832 Zozoquingu Hemimean
7 6144/6125 [11 1 -3 -2 5.3621 Sarurutrigu Porwell
7 (18 digits) [-26 -1 1 9 3.7919 Latritrizo-ayo Wadisma
7 65625/65536 [-16 1 5 1 2.3495 Lazoquinyo Horwell
7 (12 digits) [-11 2 7 -3 1.6283 Latriru-asepyo Meter
7 2401/2400 [-5 -1 -2 4 0.72120 Bizozogu Breedsma
11 99/98 [-1 2 0 -2 1 17.576 Loruru Mothwellsma
11 121/120 [-3 -1 -1 0 2 14.367 Lologu Biyatisma
11 176/175 [4 0 -2 -1 1 9.8646 Lorugugu Valinorsma
11 243/242 [-1 5 0 0 -2 7.1391 Lulu Rastma
11 385/384 [-7 -1 1 1 1 4.5026 Lozoyo Keenanisma
11 441/440 [-3 2 -1 2 -1 3.9302 Luzozogu Werckisma
11 540/539 [2 3 1 -2 -1 3.2090 Lururuyo Swetisma
11 3025/3024 [-4 -3 2 -1 2 0.57240 Loloruyoyo Lehmerisma
13 105/104 [-3 1 1 1 0 -1 16.567 Thuzoyo Animist comma
13 144/143 [4 2 0 0 -1 -1 12.064 Thulu Grossma
13 196/195 [2 -1 -1 2 0 -1 8.8554 Thuzozogu Mynucuma
  1. Ratios longer than 10 digits are presented by placeholders with informative hints

Rank-2 temperaments

Rank-2 temperaments by generators
Generator Cents MOSes Temperaments Pergen
1\31 38.71 Slender (P8, P4/13)
2\31 77.42 1L 14s, 15L 1s Valentine / lupercalia (P8, P5/9)
3\31 116.13 1L 9s, 10L 1s, 10L 11s Mercy / miracle (P8, P5/6)
4\31 154.84 1L 6s, 7L 1s,
8L 7s, 8L 15s
Nusecond / greeley (P8, P11/11)
5\31 193.55 1L 5s, 6L 1s, 6L 7s,
6L 13s, 6L 19s
Luna / didacus / hemithirds /

hemiwürschmidt / tutone

(P8, ccP4/15)
6\31 232.26 1L 4s, 5L 1s, 5L 6s,
5L 11s, 5L 16s, 5L 21s
Mothra / mosura
Quadrawell
(P8, P5/3)
7\31 270.97 1L 3s, 4L 1s, 4L 5s,
9L 4s, 9L 13s
Orson / orwell / winston (P8, P12/7)
8\31 309.68 3L 1s, 4L 3s, 4L 7s,
4L 11s, 4L 15s, 4L 19s,
4L 23s
Myna / triwell (P8, ccP5/10)
9\31 348.39 3L 1s, 3L 4s, 7L 3s,
7L 10s, 7L 17s
Mohaha / vicentino /
mohajira / migration
(P8, P5/2)
10\31 387.10 3L 1s, 3L 4s, 3L 7s,
3L 10s, 3L 13s, 3L 16s,
3L 19s, 3L 22s, 3L 25s
Würschmidt / worschmidt (P8, ccP5/8)
11\31 425.81 3L 2s, 3L 5s, 3L 8s,
3L 11s, 14L 3s
Squares / Sentinel (P8, P11/4)
12\31 464.52 3L 2s, 5L 3s,
5L 8s, 13L 5s
A-Team
Semisept
(P8, c5P4/14)
13\31 503.23 2L 3s, 5L 2s,
7L 5s, 12L 7s
Meantone / meanpop (P8, P5)
14\31 541.94 2L 3s, 2L 5s, 2L 7s,
9L 2s, 11L 9s
Casablanca
Cypress
Oracle
(P8, c5P4/12)
15\31 580.65 2L 3s, 2L 5s, 2L 7s,
2L 9s, 2L 11s, 2L 13s,
2L 15s, 2L 17s, 2L 19s,
2L 21s, 2L 23s, 2L 25s,
2L 27s
Tritonic / tritoni (P8, ccP4/5)

Music

See also: Category:31edo tracks

By Cam Taylor

By Diamond Doll (Xen-Pop)

By Johann alias Circular17

By Zhea Erose

See also

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