31edo

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Thirty-one tone equal temperament, also called 31-tET, 31-EDO, 31-et, or tricesimoprimal meantone temperament, is the scale derived by dividing the octave into 31 equally large steps. The term Tricesimoprimal was first used by Adriaan Fokker.

Theory

Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 cents. 31'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). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth zeta integral edo. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It also deals with the 11-limit fairly well, and is consistent through it, but is the optimal patent val for the rank five temperament tempering out the 13-limit comma 66/65. 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.

31edo's 12\31 generator (an approximate 21/16) supports A-Team and yields 8-note "oneirotonic" scales similar to those in 13edo but with the 9/8 and 5/4 better in tune; this temperament is also represented by 13edo, 18edo and 44edo.

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

For more encyclopedic info, see Wikipedia's article.

For a list of diatonic key signatures and major scales in 31edo (including semi- and sesqui- sharps); check this video and docs in its description.

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 ^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#, Gbb
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 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 monzo format examples
downminor zo {a, b, 0, 1} 7/6, 7/4
minor fourthward wa {a, b}, 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}, 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 edosteps 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 Ups and Downs Notation - Chords and Chord Progressions.

Just approximation

Selected just intervals

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23
Error absolute (¢) 0.0 -5.18 +0.8 -1.1 -9.4 +11.1 +11.2 +12.2 -8.9
relative (%) 0.0 -13.4 +2.0 -2.8 -24.2 +28.6 +28.9 +31.4 -23.0
fifthspan 0 +1 +4 +10 -13 +15 -5 -3 -6

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.

Direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢)
5/4, 8/5 0.783
11/9, 18/11 0.979
8/7, 7/4 1.084
7/5, 10/7 1.867
15/14, 28/15 3.314
7/6, 12/7 4.097
12/11, 11/6 4.202
16/15, 15/8 4.398
15/11, 22/15 4.985
4/3, 3/2 5.181
6/5, 5/3 5.964
14/11, 11/7 8.298
9/7, 14/9 9.278
11/8, 16/11 9.382
11/10, 20/11 10.166
13/10, 20/13 10.302
9/8, 16/9 10.362
16/13, 13/8 11.085
10/9, 9/5 11.145
14/13, 13/7 12.169
15/13, 26/15 15.483
13/12, 24/13 16.266
18/13, 13/9 17.263
13/11, 22/13 18.242
Patent val mapping
Interval, complement Error (abs, ¢)
5/4, 8/5 0.783
11/9, 18/11 0.979
8/7, 7/4 1.084
7/5, 10/7 1.867
15/14, 28/15 3.314
7/6, 12/7 4.097
12/11, 11/6 4.202
16/15, 15/8 4.398
15/11, 22/15 4.985
4/3, 3/2 5.181
6/5, 5/3 5.964
14/11, 11/7 8.298
9/7, 14/9 9.278
11/8, 16/11 9.382
11/10, 20/11 10.166
13/10, 20/13 10.302
9/8, 16/9 10.362
16/13, 13/8 11.085
10/9, 9/5 11.145
14/13, 13/7 12.169
15/13, 26/15 15.483
13/12, 24/13 16.266
13/11, 22/13 20.468
18/13, 13/9 21.447

Selected 19-limit intervals

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Individual degrees of 31edo

1\31 - 38.71¢ - Diesis or up-unison

A single step of 31-edo is about 38.71¢. Intervals around this size are called dieses (singular diesis). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the 11-limit, the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. 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. Demonstrated in SpiralProgressions.

2\31 - 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second or downminor 2nd

The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two "microtones". In meantone, it is the chromatic semitone, the interval that distinguishes major and minor intervals of the same generic interval class (e.g. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates valentine temperament - aka semi-equalized Armodue.

MOS Scales generated by 2\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
15-tone (ME or quasi-equal) 1L 14s 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3
16-tone 15L 1s 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1

3\31 - 116.13¢ - Major Semitone or Diatonic Semitone or Large Minor Second or minor 2nd

The larger and clunkier of the 31edo semitones. In meantone, it is the diatonic semitone which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a secor and generates miracle temperament. The Pythagorean apotome 2187:2048 (113.69¢) is close to 3\31 in value, but is not consistent with the mapping of the primes 2 and 3 in 31edo (in fact the apotome of 31edo is the previous degree 2\31).

MOS Scales generated by 3\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
nonatonic 1L 8s 3 3 3 3 3 3 3 3 7
decatonic (quasi-equal) 9L 1s 3 3 3 3 3 3 3 3 3 4
11-tone 10L 1s 3 3 3 3 3 3 3 3 3 3 1
21-tone (Blackjack) 11L 10s 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1

4\31 - 154.84¢ - Neutral Tone or Neutral Second or mid 2nd

Exactly one half of the minor third and twice the minor semitone. 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Although neutral seconds are typically associated with the 11-limit, 4\31 approximates the 7-limit interval 35/32 quite well, as the 5th harmonic of the 7th harmonic or vice versa, both of which are closely approximated in 31edo. And although 31 is not extremely accurate in the 11-limit, it is notable that since 11 and 3 are both flat, the interval that distinguishes them (12/11) is only about 4.5¢ off. Generates nusecond temperament.

MOS Scales generated by 4\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
heptatonic 1L 6s 4 4 4 4 4 4 7
octatonic (quasi-equal) 7L 1s 4 4 4 4 4 4 4 3
15-tone 8L 7s 1 3 1 3 1 3 1 3 1 3 1 3 1 3 3
23-tone 8L 15s 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 2

5\31 - 193.55¢ - Whole Tone or Major Second or major 2nd

A rather smallish whole tone. Sometimes called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a "meantone". Two meantones make a near-just major third. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat "active" (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps 0-18-36-45). Generates hemithirds temperament and hermiwuerschmidt temperament.

MOS Scales generated by 5\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
hexatonic (quasi-equal) 1L 5s 5 5 5 5 5 6
heptatonic 6L 1s 5 5 5 5 5 5 1
13-tone 6L 7s 4 1 4 1 4 1 4 1 4 1 4 1 1
19-tone 6L 13s 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 1
25-tone 6L 19s 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 1

6\31 - 232.26¢ - Supermajor Second or upmajor 2nd

Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 closely represents 8:7 (231.17¢). In meantone, it is a diminished third, e.g. C to Ebb. Generates mothra temperament.

MOS Scales generated by 6\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
pentatonic (quasi-equal) 1L 4s 6 6 6 6 7
hexatonic 5L 1s 6 6 6 6 6 1
11-tone 5L 6s 5 1 5 1 5 1 5 1 5 1 1
16-tone 5L 11s 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 1
21-tone 5L 16s 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 1
26-tone 5L 21s 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1

7\31 - 270.97¢ - Subminor Third or downminor 3rd

Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). In meantone temperament, it is an augmented 2nd, e.g. C to D#. Generates orwell temperament.

MOS Scales generated by 7\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
pentatonic 4L 1s 7 7 7 7 3
nonatonic (quasi-equal; Orwell[9]) 4L 5s 4 3 4 3 4 3 4 3 3
13-tone (Orwell[13]) 9L 4s 1 3 3 1 3 3 1 3 3 1 3 3 3
22-tone (Orwell[22]) 9L 13s 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 2

8\31 - 309.68¢ - Minor Third

A minor third, closer to the just 6:5 (315.64¢) than 12-edo, but still on the flat side. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates myna temperament.

MOS Scales generated by 8\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tetratonic (quasi-equal) 3L 1s 8 8 8 7
heptatonic 4L 3s 1 7 1 7 1 7 7
11-tone 4L 7s 1 1 6 1 1 6 1 1 6 1 6
15-tone 4L 11s 1 1 1 5 1 1 1 5 1 1 1 5 1 1 5
19-tone 4L 15s 1 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4
23-tone 4L 19s 1 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 3
27-tone 4L 23s 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 2

9\31 - 348.39¢ - Neutral Third or mid 3rd

A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for mohajira temperament), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a 13-limit interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more "major-like" than "minor-like"). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, which leaves the 143rd harmonic only about 2¢ wide after cancelling with the narrow 11th harmonic, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.

MOS Scales generated by 9\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tetratonic 3L 1s 9 9 9 4
heptatonic (quasi-equal) 3L 4s 5 4 5 4 5 4 4
10-tone 7L 3s 1 4 4 1 4 4 1 4 4 4
17-tone 7L 10s 1 1 3 1 3 1 1 3 1 3 1 1 3 1 3 1 3
24-tone 7L 17s 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 2

10\31 - 387.10¢ - Major Third

A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as "smooth". Generates wurshmidt/worshmidt temperaments.

MOS Scales generated by 10\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic (quasi-equal) 1L 2s 10 10 11
tetratonic 3L 1s 10 10 10 1
heptatonic 3L 4s 9 1 9 1 9 1 1
10-tone 3L 7s 8 1 1 8 1 1 8 1 1 1
13-tone 3L 10s 7 1 1 1 7 1 1 1 7 1 1 1 1
16-tone 3L 13s 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 1
19-tone 3L 16s 5 1 1 1 1 1 5 1 1 1 1 1 5 1 1 1 1 1 1
22-tone 3L 19s 4 1 1 1 1 1 1 4 1 1 1 1 1 1 4 1 1 1 1 1 1 1
25-tone 3L 22s 3 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1
28-tone 3L 25s 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1

11\31 - 425.806¢ - Supermajor Third or upmajor 3rd

11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, e.g. C to Fb. It is notable as closely approximating an interval of the 23-limit, suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular 17edo. Generates squares temperament.

MOS Scales generated by 11\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 11 11 9
pentatonic 3L 2s 2 9 2 9 9
octatonic 3L 5s 2 2 7 2 2 7 2 7
11-tone 3L 8s 2 2 2 5 2 2 2 5 2 2 5
14-tone (quasi-equal) 3L 11s 2 2 2 2 3 2 2 2 2 3 2 2 2 3
17-tone 3L 14s 2 2 2 2 2 1 2 2 2 2 2 2 1 2 2 2 1

12\31 - 464.52¢ - Narrow Fourth or Subfourth or down 4th

Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). It is also quite close to the 17-limit interval 17/13 (464.43¢), although 31edo does not offer up reasonable approximations of the 17th or 13th harmonics to help make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-limit consonances, and the only intervals in the 31edo octave that are not 15-limit consonances. Generates A-Team and semisept temperaments.

MOS Scales generated by 12\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 12 12 7
pentatonic 3L 2s 5 7 5 7 7
octatonic 5L 3s 5 5 2 5 5 2 5 2
13-tone (quasi-equal) 5L 8s 3 2 3 2 2 3 2 3 2 2 3 2 2
18-tone 13L 5s 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2

13\31 - 503.23¢ - Perfect Fourth

A slightly wide perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for meantone temperament.

MOS Scales generated by 13\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 13 13 5
pentatonic 2L 3s 8 5 8 5 5
heptatonic 5L 2s 3 5 5 3 5 5 5
12-tone (quasi-equal) 7L 5s 3 3 2 3 2 3 3 2 3 2 3 2
19-tone 12L 7s 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2

14\31 - 541.94¢ - Superfourth or up 4th

Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates casablanca and joan temperaments.

MOS Scales generated by 14\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 14 14 3
pentatonic 2L 3s 11 3 11 3 3
heptatonic 2L 5s 8 3 3 8 3 3 3
nonatonic 2L 7s 5 3 3 3 5 3 3 3 3
11-tone (quasi-equal) 9L 2s 2 3 3 3 3 2 3 3 3 3 3
20-tone 11L 9s 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1

15\31 - 580.65¢ - Small Tritone or Augmented 4th or Subdiminished Fifth or downdim 5th

In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates tritonic temperament.

MOS Scales generated by 15\31

number of tones MOS class 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
tritonic 2L 1s 15 15 1
pentatonic 2L 3s 14 1 14 1 1
heptatonic 2L 5s 13 1 1 13 1 1 1
nonatonic 2L 7s 12 1 1 1 12 1 1 1 1
11-tone 2L 9s 11 1 1 1 1 11 1 1 1 1 1
13-tone 2L 11s 10 1 1 1 1 1 10 1 1 1 1 1 1
15-tone 2L 13s 9 1 1 1 1 1 1 9 1 1 1 1 1 1 1
17-tone 2L 15s 8 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1
19-tone 2L 17s 7 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 1 1
21-tone 2L 19s 6 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1
23-tone 2L 21s 5 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1
25-tone 2L 23s 4 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1
27-tone 2L 25s 3 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1
29-tone 2L 27s 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1

16\31 - Large Tritone or dim 5th

Etc.

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 31et.

3-limit 5-limit 7-limit 11-limit 13-limit
Octave stretch (¢) +1.63 +0.98 +0.83 +1.21 +0.50
Error absolute (¢) 1.64 1.63 1.43 1.49 2.07
relative (%) 4.22 4.20 3.70 3.84 5.35
  • 31et has a lower relative error than any previous ETs in all harmonic limits from 7 to 31.

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

Commas

31 EDO tempers out the following commas. (Note: This assumes the val < 31 49 72 87 107 115 |, comma values rounded to 5 significant digits.)

Ratio Monzo Cents Color Name Name 1 Name 2 Name 3
34171875/33554432 | -25 7 6 > 31.567 Lala-tribiyo Ampersand's Comma
81/80 | -4 4 -1 > 21.506 Gu Syntonic Comma Didymos Comma Meantone Comma
393216/390625 | 17 1 -8 > 11.445 Saquadbigu Würschmidt Comma
2109375/2097152 | -21 3 7 > 10.061 Lasepyo Semicomma Fokker Comma
6719816/6714445 | 38 -2 -15 > 1.3843 Sasa-quintrigu Hemithirds Comma
9859966/9733137 | -10 7 8 -7 > 22.413 Lasepru-aquadbiyo Blackjackisma
64827/64000 | -9 3 -3 4 > 22.227 Laquadzo-atrigu Squalentine
2430/2401 | 1 5 1 -4 > 20.785 Quadru-ayo Nuwell
50421/50000 | -4 1 -5 5 > 14.516 Quinzogu Trimyna
126/125 | 1 2 -3 1 > 13.795 Zotrigu Septimal Semicomma Starling Comma
1728/1715 | 6 3 -1 -3 > 13.074 Trizo-agu Orwellisma Orwell Comma
1029/1024 | -10 1 0 3 > 8.4327 Latrizo Gamelisma
225/224 | -5 2 2 -1 > 7.7115 Ruyoyo Septimal Kleisma Marvel Comma
16875/16807 | 0 3 4 -5 > 6.9903 Quinru-aquadyo Mirkwai
3136/3125 | 6 0 -5 2 > 6.0832 Zozoquingu Hemimean
6144/6125 | 11 1 -3 -2 > 5.3621 Sarurutrigu Porwell
1065875/1063543 | -26 -1 1 9 > 3.7919 Latritrizo-ayo Wadisma
65625/65536 | -16 1 5 1 > 2.3495 Lazoquinyo Horwell
703125/702464 | -11 2 7 -3 > 1.6283 Latriru-asepyo Meter
2401/2400 | -5 -1 -2 4 > 0.72120 Bizozogu Breedsma
99/98 | -1 2 0 -2 1 > 17.576 Loruru Mothwellsma
121/120 | -3 -1 -1 0 2 > 14.367 Lologu Biyatisma
176/175 | 4 0 -2 -1 1 > 9.8646 Lorugugu Valinorsma
243/242 | -1 5 0 0 -2 > 7.1391 Lulu Rastma
385/384 | -7 -1 1 1 1 > 4.5026 Lozoyo Keenanisma
441/440 | -3 2 -1 2 -1 > 3.9302 Luzozogu Werckisma
540/539 | 2 3 1 -2 -1 > 3.2090 Lururuyo Swetisma
3025/3024 | -4 -3 2 -1 2 > 0.57240 Loloruyoyo Lehmerisma

Linear temperaments

List of 31et rank two temperaments by badness

List of edo-distinct 31et rank two temperaments

Generator Cents Temperaments Pergen
1\31 38.71 Slender (P8, P4/13)
2\31 77.42 Valentine/Lupercalia (P8, P5/9)
3\31 116.13 Miracle (P8, P5/6)
4\31 154.84 Nusecond (P8, P11/11)
5\31 193.55 Luna/Hemithirds/Hemiwürschmidt (P8, WWP4/15)
6\31 232.26 Mothra/Mosura (P8, P5/3)
7\31 270.97 Orwell/Orson/Winston (P8, P12/7)
8\31 309.68 Myna (P8, WWP5/10)
9\31 348.39 Vicentino/Mohajira/Migration (P8, P5/2)
10\31 387.10 Würschmidt/Worschmidt (P8, WWP5/8)
11\31 425.81 Squares/Sentinel (P8, P11/4)
12\31 464.52 A-Team/Semisept (P8, M9/3)
13\31 503.23 Meantone/Meanpop (P8, P5)
14\31 541.94 Casablanca/Cypress/Oracle (P8, W5P4/12)
15\31 580.65 Tritonic/Tritoni (P8, WWP4/5)

Scales

Harmonic Scale

31edo approximates Mode 8 of the harmonic series O.K., 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 O.K., 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.

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.
  • 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 Modes

A large open list of modes (subsets) from 31edo that people have named: 31edo modes. Strictly proper 7-note 31edo scales in the sense of David Rothenberg. Interesting (to somebody) 9-note 31edo scales. See also 31edo MOS scales. Some of the popular ones:

  • 31-tone major: 5 5 3 5 5 5 3
  • Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3
  • Harmonic scale 8: 5 5 4 4 4 3 3 3
  • the Euler-Fokker genera (technically JI but representable in 31)
Some 31 tone equal modes
2 3 3 2 3 2 3 2 3 3 2 3 Meantone Chromatic (53/220-comma)
5 5 3 5 5 5 3 Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian
5 3 5 5 3 5 5 Thirty-one tone Natural Minor, Intense Diatonic Hypodorian, Aeolian
5 3 5 5 5 5 3 Thirty-one tone Melodic Minor
5 3 5 5 3 7 3 Thirty-one tone Harmonic Minor
5 5 3 5 3 7 3 Thirty-one tone Harmonic Major
5 5 3 5 3 5 5 Thirty-one tone Major-Minor
5 8 5 13 Genus primum
10 3 5 5 5 3 Genus secundum
8 2 8 3 7 3 Genus tertium
10 10 10 1 Genus quartum
5 7 6 7 5 1 Genus quintum
4 6 2 6 4 3 3 3 Genus sextum
4 6 5 6 4 6 Genus septimum
6 6 6 1 6 6 Genus octavum
4 6 9 6 4 2 Genus nonum
13 6 6 6 Genus decimum
5 5 3 5 5 3 2 3 Genus diatonicum
3 5 2 3 5 3 2 5 3 Genus chromaticum
5 5 2 1 5 5 2 3 3 Genus diatonicum cum septimis
3 4 3 3 2 1 4 1 4 1 2 3 Genus enharmonicum vocale
2 2 4 2 2 3 3 3 1 3 3 3 Genus enharmonicum instrumentale
3 2 3 2 3 2 3 3 2 3 2 3 Genus diatonico-chromaticum
5 2 1 2 5 3 2 1 4 1 2 3 Genus bichromaticum
4 4 5 4 4 5 5 Neutral Diatonic Mixolydian
4 5 4 4 5 5 4 Neutral Diatonic Lydian
5 4 4 5 5 4 4 Neutral Diatonic Phrygian
4 4 5 5 4 4 5 Neutral Diatonic Dorian
4 5 5 4 4 5 4 Neutral Diatonic Hypolydian
5 5 4 4 5 4 4 Neutral Diatonic Hypophrygian
5 4 4 5 4 4 5 Neutral Diatonic Hypodorian
4 5 4 4 5 4 5 Neutral Mixolydian
5 4 4 5 4 5 4 Neutral Lydian
4 4 5 4 5 4 5 Neutral Phrygian
4 5 4 5 4 5 4 Neutral Dorian
5 4 5 4 5 4 4 Neutral Hypolydian
4 5 4 5 4 4 5 Neutral Hypophrygian
5 4 5 4 4 5 4 Neutral Hypodorian
2 2 9 2 2 9 5 Hemiolic Chromatic Mixolydian
2 9 2 2 9 5 2 Hemiolic Chromatic Lydian
9 2 2 9 5 2 2 Hemiolic Chromatic Phrygian
2 2 9 5 2 2 9 Hemiolic Chromatic Dorian
2 9 5 2 2 9 2 Hemiolic Chromatic Hypolydian
9 5 2 2 9 2 2 Hemiolic Chromatic Hypophrygian
5 2 2 9 2 2 9 Hemiolic Chromatic Hypodorian
2 3 8 2 3 8 5 Ratio 2:3 Chromatic Mixolydian
3 8 2 3 8 5 2 Ratio 2:3 Chromatic Lydian
8 2 3 8 5 2 3 Ratio 2:3 Chromatic Phrygian
2 3 8 5 2 3 8 Ratio 2:3 Chromatic Dorian
3 8 5 2 3 8 2 Ratio 2:3 Chromatic Hypolydian
8 5 2 3 8 2 3 Ratio 2:3 Chromatic Hypophrygian
5 2 3 8 2 3 8 Ratio 2:3 Chromatic Hypodorian
3 5 5 3 5 5 5 Intense Diatonic Mixolydian, M.Locrian
5 3 5 5 5 3 5 Intense Diatonic Phrygian, M.Dorian
3 5 5 5 3 5 5 Intense Diatonic Dorian, M.Phrygian
5 5 5 3 5 5 3 Intense Diatonic Hypolydian, M.Lydian
5 5 3 5 5 3 5 Intense Diatonic Hypophrygian, M.Mixolydian
2 5 6 2 5 6 5 Soft Diatonic Mixolydian
5 6 2 5 6 5 2 Soft Diatonic Lydian
6 2 5 6 5 2 5 Soft Diatonic Phrygian
2 5 6 5 2 5 6 Soft Diatonic Dorian
5 6 5 2 5 6 2 Soft Diatonic Hypolydian
6 5 2 5 6 2 5 Soft Diatonic Hypophrygian
5 2 5 6 2 5 6 Soft Diatonic Hypodorian
1 2 10 1 2 10 5 Enharmonic Mixolydian
2 10 1 2 10 5 1 Enharmonic Lydian
10 1 2 10 5 1 2 Enharmonic Phrygian
1 2 10 5 1 2 10 Enharmonic Dorian
2 10 5 1 2 10 1 Enharmonic Hypolydian
10 5 1 2 10 1 2 Enharmonic Hypophrygian
5 1 2 10 1 2 10 Enharmonic Hypodorian
6 6 7 6 6 Quasi-equal Pentatonic
3 2 2 3 3 2 3 3 2 2 3 3 Fokker 12-tone
5 3 5 3 5 2 5 3 Modus conjunctus
3 5 2 5 3 5 3 5 Octatonic
3 3 4 3 5 3 4 3 3 Hahn symmetric pentachordal
3 4 3 3 5 3 4 3 3 Hahn pentachordal
4 4 2 5 3 3 4 3 3 Hahn Nonatonic
5 1 5 1 5 1 5 1 5 1 1 de Vries 11-tone
4 1 4 4 4 1 4 4 1 4 Breed 10-tone
4 2 4 2 4 2 4 3 3 3 Lumma Decatonic
5 3 3 3 3 5 3 3 3 Rothenberg Generalized Diatonic
5 2 6 5 2 5 6 "Septimal" Natural Minor
4 3 4 3 4 3 4 3 3 Thirty-one tone Orwell
2 5 2 2 5 2 2 2 5 2 2 Secor Sentinel

Music

An alphabetical list of Tricesimoprimal Compositions.

Aire #2 in 31-equal temperament by Jon Lyle Smith [dead link]

I Stand Hopeless Before the Gray Sea by Chuckles McGee

Chaconna en G=, La Padana, ou la septimala (‘The Padanian, or the septimal’) by Claudi Meneghin

earwig by Andrew Heathwaite

Fanfare and Toccata by Juhani Nuorvala

by Johann alias circular17: Curieuse planète, Heal, Wave from the past, Deep but not too much.

Orphanage of the Dutch Music IX: Sonate no. II in the 31-tone temperament - YouTube

Enharmonic melody for guitar by Cam Taylor

What use is a boy by Cam Taylor

Back to 31: Hyperchromatic progression on C^ by Cam Taylor

"Lively Up Yourself" cover featuring Paul Erlich on 31edo guitar

Q: IS ART REVOLUTIONARY? A: LOL NO by Diamond Doll (xen-pop)

Cool Guitar Girl by Diamond Doll (xen-pop)

Glitterdance by Zhea Erose

divinate by Zhea Erose

Some songs by Benyamin Linus

See Also

Pedagogy

The MicroPedagogyCollective is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two ThirtyOneToneSinginCamps as well.

See also: 31edo solfege, Tricesimoprimal Tetrachordal Tesseract, Pentachords of 31edo.

Books

External image: http://ronsword.com/images/TSG_sm.jpg [dead link]

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

Sword, Ronald. "Tricesimoprimal Scales for Guitar." IAAA Press, UK-USA. First Ed: March 2009. [dead link] - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.

Articles

Software