31edo

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Compact notation based on extended meantone notation: 

C  C^ C# Db Dv D  D^ D# Eb Ev E  E^ Fv F  F^ F# Gb Gv G  G^ G# Ab Av A  A^ A# Bb Bv B  B^ Cv

To round midi note a (0—127, with 60 being C4 and +12 being octave) with pitch wheel b (0—16383, with 8192 being center) to 31edo, the following formula could be used: ((a*4096L+b+32768L)*31+24575L)/49152L-26. ((a*4096L+b is the conversion of a and b to a uniform 49152edo space in a long integer, +32768L) is the subtraction of (-10*4096L+8192) to center the space on D note, *31+24575L)/49152L is the round to nearest to 31edo (with one less than half of 49152 being used to get G♯ note by default instead of A♭) and -26 adds the D-2 offset in 31edo to restore C−1 note on 0. The default 12edo range of 0—127 gets converted to 0—328 range (1536/1, or nine octaves and one tritave, in 12edo is 12×9+19=127 and in 31edo is 31×9+49=328), therefore extending 128 notes to 329 notes for C−1 to G9 range. The E♭—B♭—F—C—G—D—A—E—B—F♯—C♯—G♯ chain of fifths is preserved. Including the pitch wheel (−2 to 2 semitones of 12edo), the output range is −5—333, which is B♭−2 to A9 range of 339 notes.


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

For more encyclopedic info, see Wikipedia's article.

Linear temperaments[edit]

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 Orson/Orwell/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 Semisept (P8, W5P4/14)
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)

Intervals[edit]

Selected just intervals by error[edit]

The following table shows how some prominent just intervals are represented in 31edo (ordered by absolute error).

Best direct mapping, even if inconsistent[edit]

Interval, complement Error (abs., in cents)
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[edit]

Interval, complement Error (abs., in cents)
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

1\31 octave - approx. 38.71¢ - Diesis or up-unison[edit]

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 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second or downminor 2nd[edit]

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 (eg. 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:[edit]

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 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Minor Second or minor 2nd[edit]

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.

MOS Scales generated by 3\31:[edit]

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 octave - approx. 154.84¢ - Neutral Tone or Neutral Second or mid 2nd[edit]

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:[edit]

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 octave - approx. 193.55¢ - Whole Tone or Major Second or major 2nd[edit]

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:[edit]

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 octave - approx. 232.26¢ - Supermajor Second or upmajor 2nd[edit]

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, eg. C to Ebb. Generates mothra temperament.

MOS Scales generated by 6\31:[edit]

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 octave - approx. 270.97¢ - Subminor Third or downminor 3rd[edit]

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, eg. C to D#. Generates orwell temperament.

MOS Scales generated by 7\31:[edit]

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 octave - approx. 309.68¢ - Minor Third[edit]

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:[edit]

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 octave - approx. 348.39¢ - Neutral Third or mid 3rd[edit]

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:[edit]

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 octave - approx. 387.10¢ - Major Third[edit]

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:[edit]

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 octave - approx. 425.806¢ - Supermajor Third or upmajor 3rd[edit]

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, eg. 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:[edit]

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 octave - approx. 464.52¢ - Narrow Fourth or Subfourth or down 4th[edit]

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 semisept temperament.

MOS Scales generated by 12\31:[edit]

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 octave - approx. 503.23¢ - Perfect Fourth[edit]

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:[edit]

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 octave - approx. 541.94¢ - Superfourth or up 4th[edit]

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 temperament.

MOS Scales generated by 14\31:[edit]

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 octave - approx. 580.65¢ - Small Tritone or Augmented 4th or Subdiminished Fifth or downdim 5th[edit]

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:[edit]

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[edit]

Etc.

Notation[edit]

31edo can be notated with ups and downs notation like so:

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 31
P1 ^P1

d2

A1

vm2

m2 ~2 M2 ^M2

d3

A2

vm3

m3 ~3 M3 ^M3

d4

A3

v4

P4 ^4 A4 d5 v5 P5 ^5

d6

A5

vm6

m6 ~6 M6 ^M6

d7

A6

vm7

m7 ~7 M7 ^M7

d8

A7

v8

P8

^ = up, v = down, ~ = mid

vm2 = downminor 2nd, ~2 = mid 2nd, ^M2 = upmajor 2nd, etc.

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 31
C C^ C#

Dbv

C#^

Db

Dv D D^ D#

Ebv

D#^

Eb

Ev E E^

Fb

E#

Fv

F F^ F#

Gbv

F#^

Gb

Gv G G^ G#

Abv

G#^

Ab

Av A A^ A#

Bbv

A#^

Bb

Bv B B^

Cb

B#

Cv

C

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 lova {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. Here are the zo, gu, lova, yo and ru triads:

color of the 3rd JI chord notes as edosteps notes of C chord written name spoken name
zo 6:7:9 0-7-18 C Ebv G C.vm C downminor
gu 10:12:15 0-8-18 C Eb G Cm C minor
lova 18:22:27 0-9-18 C Ev G C~ C mid
yo 4:5:6 0-10-18 C E G C C major or C
ru 14:18:27 0-11-18 C E^ G C.^ C upmajor or C dot up

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Harmonic Scale[edit]

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.
  • 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 subset, 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

Commas[edit]

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 Value (Cents) Name 1 Name 2 Name 3
34171875/33554432 | -25 7 6 > 31.567 Ampersand's Comma
81/80 | -4 4 -1 > 21.506 Syntonic Comma Didymos Comma Meantone Comma
393216/390625 | 17 1 -8 > 11.445 Würschmidt Comma
2109375/2097152 | -21 3 7 > 10.061 Semicomma Fokker Comma
274877906944/274658203125 | 38 -2 -15 > 1.3843 Hemithirds Comma
854296875/843308032 | -10 7 8 -7 > 22.413 Blackjackisma
64827/64000 | -9 3 -3 4 > 22.227 Squalentine
2430/2401 | 1 5 1 -4 > 20.785 Nuwell
50421/50000 | -4 1 -5 5 > 14.516 Trimyna
126/125 | 1 2 -3 1 > 13.795 Septimal Semicomma Starling Comma
1728/1715 | 6 3 -1 -3 > 13.074 Orwellisma Orwell Comma
1029/1024 | -10 1 0 3 > 8.4327 Gamelisma
225/224 | -5 2 2 -1 > 7.7115 Septimal Kleisma Marvel Comma
16875/16807 | 0 3 4 -5 > 6.9903 Mirkwai
3136/3125 | 6 0 -5 2 > 6.0832 Hemimean
6144/6125 | 11 1 -3 -2 > 5.3621 Porwell
201768035/201326592 | -26 -1 1 9 > 3.7919 Wadisma
65625/65536 | -16 1 5 1 > 2.3495 Horwell
703125/702464 | -11 2 7 -3 > 1.6283 Meter
2401/2400 | -5 -1 -2 4 > 0.72120 Breedsma
99/98 | -1 2 0 -2 1 > 17.576 Mothwellsma
121/120 | -3 -1 -1 0 2 > 14.367 Biyatisma
176/175 | 4 0 -2 -1 1 > 9.8646 Valinorsma
243/242 | -1 5 0 0 -2 > 7.1391 Rastma
385/384 | -7 -1 1 1 1 > 4.5026 Keenanisma
441/440 | -3 2 -1 2 -1 > 3.9302 Werckisma
540/539 | 2 3 1 -2 -1 > 3.2090 Swetisma
3025/3024 | -4 -3 2 -1 2 > 0.57240 Lehmerisma

Modes[edit]

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:[edit]

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 in 31-edo[edit]

An alphabetical list of Tricesimoprimal Compositions.

Aire #2 in 31-equal temperament by Jon Lyle Smith

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

Thirty-one tone pedagogy[edit]

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.

Practical Theory / Books[edit]

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.

Other Articles[edit]

File:31-edo.svg

31-edo.svg

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