32edo: Difference between revisions
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== Theory == | == Theory == | ||
32edo is generally the first power-of-2 edo which can be considered to handle [[limit|low-limit]] just intonation at all. It has unambiguous mappings for [[prime]]s up to the [[11-limit]], although [[6/5]] and Pythagorean intervals are especially poorly approximated if going by the [[patent val]] instead of using [[direct approximation|inconsistent approximations]]. Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[mos]] theory are of limited usefulness in the system (though it has an [[ultrasoft]] [[smitonic]] with {{nowrap|L/s {{=}} 5/4}}). 32edo's 5:2:1 [[blackdye]] scale {{nowrap|(1 5 2 5 1 5 2 5 1 5)}}, which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale. | 32edo is generally the first power-of-2 edo which can be considered to handle [[limit|low-limit]] just intonation at all. It has unambiguous mappings for [[prime]]s up to the [[11-limit]], although [[6/5]] and Pythagorean intervals are especially poorly approximated if going by the [[patent val]] instead of using [[direct approximation|inconsistent approximations]]. Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[mos]] theory are of limited usefulness in the system (though it has an [[ultrasoft]] [[smitonic]] with {{nowrap|L/s {{=}} 5/4}}). 32edo's 5:2:1 [[blackdye]] scale {{nowrap|(1 5 2 5 1 5 2 5 1 5)}}, which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale. | ||
=== Odd harmonics === | |||
{{Harmonics in equal|32}} | |||
=== As a tuning of other temperaments === | === As a tuning of other temperaments === | ||
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The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[oceanfront]]) [[5L 2s|diatonic scale]], with a [[diatonic semitone]] of 5 steps and a [[chromatic semitone]] of only 1. The diatonic [[major third]] (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[9/7]] and [[13/10]], and the minor third is 262.5¢, approximating [[7/6]]. In addition to the sharp fifth, there is an alternative [[mavila|mavila-like]] flat fifth of 675{{c}} (inherited from [[16edo]]), but it is much more inaccurate than the sharp fifth, so 32edo cannot truly be considered a [[dual-fifth]] system. | The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[oceanfront]]) [[5L 2s|diatonic scale]], with a [[diatonic semitone]] of 5 steps and a [[chromatic semitone]] of only 1. The diatonic [[major third]] (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[9/7]] and [[13/10]], and the minor third is 262.5¢, approximating [[7/6]]. In addition to the sharp fifth, there is an alternative [[mavila|mavila-like]] flat fifth of 675{{c}} (inherited from [[16edo]]), but it is much more inaccurate than the sharp fifth, so 32edo cannot truly be considered a [[dual-fifth]] system. | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Revision as of 08:48, 29 October 2026
| ← 31edo | 32edo | 33edo → |
32 equal divisions of the octave (abbreviated 32edo or 32ed2), also called 32-tone equal temperament (32tet) or 32 equal temperament (32et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 32 equal parts of exactly 37.5 ¢ each. Each step represents a frequency ratio of 21/32, or the 32nd root of 2.
Theory
32edo is generally the first power-of-2 edo which can be considered to handle low-limit just intonation at all. It has unambiguous mappings for primes up to the 11-limit, although 6/5 and Pythagorean intervals are especially poorly approximated if going by the patent val instead of using inconsistent approximations. Since 32edo is poor at approximating primes and it is a high power of 2, both traditional RTT and temperament-agnostic mos theory are of limited usefulness in the system (though it has an ultrasoft smitonic with L/s = 5/4). 32edo's 5:2:1 blackdye scale (1 5 2 5 1 5 2 5 1 5), which is melodically comparable to 31edo's 5:2:1 diasem, is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and 40/27 in 5-limit blackdye, making 5:2:1 blackdye a dual-fifth scale.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.5 | -11.3 | +6.2 | -16.4 | +11.2 | -15.5 | -0.8 | +7.5 | +2.5 | +16.7 | +9.2 |
| Relative (%) | +28.1 | -30.2 | +16.5 | -43.8 | +29.8 | -41.4 | -2.0 | +20.1 | +6.6 | +44.6 | +24.6 | |
| Steps (reduced) |
51 (19) |
74 (10) |
90 (26) |
101 (5) |
111 (15) |
118 (22) |
125 (29) |
131 (3) |
136 (8) |
141 (13) |
145 (17) | |
As a tuning of other temperaments
32edo provides a good tuning for the sixix temperament, which tempers out the 5-limit sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents. Sixix temperament is generated by an approximate 6/5, which is tuned very sharp so that 5 of them octave-reduce to 4/3. Petr Pařízek's preferred generator for sixix is (128/15)(1/11), which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. The 9\32 generator can also be used for the much more inaccurate mohavila temperament which treats the 675-cent fifth from 16edo as 3/2 (basically it's mavila's response to neutral third temperaments).
It also tempers out 2048/2025 in the 5-limit, and 50/49 with 64/63 in the 7-limit, which means it supports pajara, with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of 27edo; this fifth is in fact very close to the minimax tuning of the pajara extension pajaro, using the 32f val. In the 11-limit it provides the optimal patent val for kangaroo temperament, tempering out 55/54, 64/63, and 245/242. This is generated by a flat 16/11 such that 3 of them reduce to 3/2 and 6 of them reduce to 8/7. Its 15-note scale could potentially be used as a detempering of 15edo.
The sharp fifth of 32edo can be used to generate a very unequal archy (specifically oceanfront) diatonic scale, with a diatonic semitone of 5 steps and a chromatic semitone of only 1. The diatonic major third (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating 9/7 and 13/10, and the minor third is 262.5¢, approximating 7/6. In addition to the sharp fifth, there is an alternative mavila-like flat fifth of 675 ¢ (inherited from 16edo), but it is much more inaccurate than the sharp fifth, so 32edo cannot truly be considered a dual-fifth system.
Subsets and supersets
Since 32 is a power of two and factors as 25, 32edo contains subset edos 2, 4, 8, and 16.
See also 32nd-octave temperaments.
Intervals
| Degree | Cents | Ups and downs notation | 13-limit Ratios | Other | ||
|---|---|---|---|---|---|---|
| 0 | 0.0 | P1 | perfect unison | D | 1/1 | |
| 1 | 37.5 | ^1, m2 | up unison, minor 2nd | ^D, Eb | 49/48, 50/49, 45/44 | 46/45, 52/51, 51/50 |
| 2 | 75.0 | ^m2 | upminor 2nd | ^Eb | 22/21, 25/24 | 24/23, 23/22 |
| 3 | 112.5 | ^^m2 | dupminor 2nd | ^^Eb | 16/15 | 49/46 |
| 4 | 150.0 | vvM2 | dudmajor 2nd | vvE | 12/11, 49/45 | 25/23 |
| 5 | 187.5 | A1, vM2 | aug 1sn, downmajor 2nd | D#, vE | 10/9, 39/35 | 19/17 |
| 6 | 225.0 | M2 | major 2nd | E | 8/7, 25/22 | 57/50 |
| 7 | 262.5 | m3 | minor 3rd | F | 7/6, 64/55 | 57/49 |
| 8 | 300.0 | ^m3 | upminor 3rd | ^F | 6/5, 32/27 | 19/16 |
| 9 | 337.5 | ^^m3 | dupminor 3rd | ^^F | 11/9, 39/32, 63/52 | 17/14, 28/23 |
| 10 | 375.0 | vvM3 | dudmajor 3rd | vvF# | 5/4, 26/21, 56/45, 96/77 | 36/29 |
| 11 | 412.5 | vM3 | downmajor 3rd | vF# | 14/11, 33/26, 80/63 | 19/15 |
| 12 | 450.0 | M3 | major 3rd | F# | 13/10, 35/27, 64/49 | 22/17, 57/44 |
| 13 | 487.5 | P4 | perfect 4th | G | 4/3, 33/25, 160/121 | 45/34, 85/64 |
| 14 | 525.0 | ^4 | up 4th | ^G | 27/20, 110/81 | 19/14, 23/17 |
| 15 | 562.5 | ^^4, ^d5 | dup 4th, updim 5th | ^^G, ^Ab | 18/13, 11/8 | |
| 16 | 600.0 | vvA4, ^^d5 | dudaug 4th, dupdim 5th | vvG#, ^^Ab | 7/5, 10/7, 99/70, 140/99 | 17/12, 12/17 |
| 17 | 637.5 | vA4, vv5 | downaug 4th, dud 5th | vG#, vvA | 13/9, 16/11 | |
| 18 | 675.0 | v5 | down 5th | vA | 40/27, 81/55 | 28/19, 34/23 |
| 19 | 712.5 | P5 | perfect 5th | A | 3/2, 50/33, 121/80 | 68/45, 128/85 |
| 20 | 750.0 | m6 | minor 6th | Bb | 20/13, 54/35, 49/32 | 17/11, 88/57 |
| 21 | 787.5 | ^m6 | upminor 6th | ^Bb | 11/7, 52/33, 63/40 | 30/19 |
| 22 | 825.0 | ^^m6 | dupminor 6th | ^^Bb | 8/5, 21/13, 45/28, 77/48 | 29/18 |
| 23 | 862.5 | vvM6 | dudmajor 6th | vvB | 18/11, 64/39, 104/63 | 28/17, 23/14 |
| 24 | 900.0 | vM6 | downmajor 6th | vB | 5/3, 27/16 | 32/19 |
| 25 | 937.5 | M6 | major 6th | B | 12/7, 55/32 | 98/57 |
| 26 | 975.0 | m7 | minor 7th | C | 7/4, 44/25 | 100/57 |
| 27 | 1012.5 | ^m7 | upminor 7th | ^C | 9/5, 70/39 | 34/19 |
| 28 | 1050.0 | ^^m7 | dupminor 7th | ^^C | 11/6, 90/49 | 46/25 |
| 29 | 1087.5 | vvM7 | dudmajor 7th | vvC# | 15/8 | 92/49 |
| 30 | 1125.0 | vM7 | downmajor 7th | vC# | 21/11, 48/25 | 23/12, 44/23 |
| 31 | 1162.5 | M7, v8 | major 7th, down 8ve | C#, vD | 96/49, 49/25, 88/45 | 45/23, 51/26, 100/51 |
| 32 | 1200.0 | P8 | 8ve | D | 2/1 | |
Notation
Stein–Zimmermann–Gould notation
Stein–Zimmermann–Gould notation uses sharps and flats with arrows:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | | | | | | | | | | | | | |
| Flat symbol | | | | | | | | | | | | |
If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using triple arrows.
Kite's ups and downs notation
32edo can also be notated with Kite's ups and downs, spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
Sagittal notation
This notation uses the same sagittal sequence as 25edo, and is a subset of the notation for 64b-edo.
Evo flavor

Revo flavor

Approximation to JI
The following tables show how 15-odd-limit intervals are represented in 32edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 11/6, 12/11 | 0.637 | 1.7 |
| 15/8, 16/15 | 0.769 | 2.0 |
| 13/9, 18/13 | 0.882 | 2.4 |
| 13/10, 20/13 | 4.214 | 11.2 |
| 7/6, 12/7 | 4.371 | 11.7 |
| 11/7, 14/11 | 5.008 | 13.4 |
| 9/5, 10/9 | 5.096 | 13.6 |
| 7/4, 8/7 | 6.174 | 16.5 |
| 15/14, 28/15 | 6.943 | 18.5 |
| 11/9, 18/11 | 9.908 | 26.4 |
| 3/2, 4/3 | 10.545 | 28.1 |
| 13/11, 22/13 | 10.790 | 28.8 |
| 11/8, 16/11 | 11.182 | 29.8 |
| 5/4, 8/5 | 11.314 | 30.2 |
| 13/12, 24/13 | 11.427 | 30.5 |
| 15/11, 22/15 | 11.951 | 31.9 |
| 15/13, 26/15 | 14.759 | 39.4 |
| 9/7, 14/9 | 14.916 | 39.8 |
| 11/10, 20/11 | 15.004 | 40.0 |
| 13/8, 16/13 | 15.528 | 41.4 |
| 5/3, 6/5 | 15.641 | 41.7 |
| 13/7, 14/13 | 15.798 | 42.1 |
| 9/8, 16/9 | 16.410 | 43.8 |
| 7/5, 10/7 | 17.488 | 46.6 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 11/6, 12/11 | 0.637 | 1.7 |
| 15/8, 16/15 | 0.769 | 2.0 |
| 13/10, 20/13 | 4.214 | 11.2 |
| 7/6, 12/7 | 4.371 | 11.7 |
| 11/7, 14/11 | 5.008 | 13.4 |
| 7/4, 8/7 | 6.174 | 16.5 |
| 15/14, 28/15 | 6.943 | 18.5 |
| 11/9, 18/11 | 9.908 | 26.4 |
| 3/2, 4/3 | 10.545 | 28.1 |
| 11/8, 16/11 | 11.182 | 29.8 |
| 5/4, 8/5 | 11.314 | 30.2 |
| 15/11, 22/15 | 11.951 | 31.9 |
| 15/13, 26/15 | 14.759 | 39.4 |
| 9/7, 14/9 | 14.916 | 39.8 |
| 13/8, 16/13 | 15.528 | 41.4 |
| 7/5, 10/7 | 17.488 | 46.6 |
| 9/8, 16/9 | 21.090 | 56.2 |
| 13/7, 14/13 | 21.702 | 57.9 |
| 5/3, 6/5 | 21.859 | 58.3 |
| 11/10, 20/11 | 22.496 | 60.0 |
| 13/12, 24/13 | 26.073 | 69.5 |
| 13/11, 22/13 | 26.710 | 71.2 |
| 9/5, 10/9 | 32.404 | 86.4 |
| 13/9, 18/13 | 36.618 | 97.6 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [51 -32⟩ | [⟨32 51]] | -3.327 | 3.32 | 8.87 |
| 2.3.7 | 64/63, 46118408/43046721 | [⟨32 51 90]] | -2.950 | 2.76 | 7.38 |
| 2.3.5 | 648/625, 20480/19683 | [⟨32 51 75]] (32c) | -5.965 | 4.61 | 12.3 |
| 2.3.5.7 | 64/63, 245/243, 392/375 | [⟨32 51 75 90]] (32c) | -5.027 | 4.31 | 11.5 |
| 2.3.5 | 2048/2025, 3125/2916 | [⟨32 51 74]] (32) | +0.177 | 4.72 | 12.6 |
| 2.3.5.7 | 50/49, 64/63, 3125/2916 | [⟨32 51 75 90]] (32) | -1.008 | 4.15 | 11.1 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 1\32 | 37.5 | 49/48 | Slender (32) |
| 1 | 9\32 | 262.5 | 7/6 | Septimin (32f) |
| 1 | 9\32 | 337.5 | 6/5 | Sixix (32f) |
| 1 | 13\32 | 487.5 | 4/3 | Superpyth (32c, 7-limit) / ultrapyth (32) / quasiultra (32) |
| 1 | 15\32 | 562.5 | 7/5 | Progress (32cf) |
| 2 | 13\32 | 487.5 | 4/3 | Pajara (32, 7-limit) |
| 8 | 14\33 (1\32) |
487.5 (37.5) |
4/3 (36/35) |
Octonion (32cf) |
| 16 | 14\33 (1\32) |
487.5 (37.5) |
4/3 (45/44) |
Sedecic (32) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Delta-rational harmony
The tables below show chords that approximate 3-integer-limit delta-rational chords with least-squares error less than 0.0015.
| Steps | Delta signature | Least-squares error |
|---|---|---|
| 0,1,2 | +1+1 | 0.00023 |
| 0,1,3 | +1+2 | 0.00051 |
| 0,1,4 | +1+3 | 0.00083 |
| 0,2,3 | +2+1 | 0.00041 |
| 0,2,4 | +1+1 | 0.00092 |
| 0,3,4 | +3+1 | 0.00060 |
| 0,3,11 | +1+3 | 0.00014 |
| 0,4,11 | +1+2 | 0.00087 |
| 0,5,8 | +3+2 | 0.00076 |
| 0,6,16 | +1+2 | 0.00076 |
| 0,8,26 | +1+3 | 0.00016 |
| 0,9,23 | +1+2 | 0.00000 |
| 0,12,17 | +2+1 | 0.00004 |
| 0,13,20 | +3+2 | 0.00008 |
| 0,15,21 | +2+1 | 0.00007 |
| 0,18,27 | +3+2 | 0.00000 |
| 0,22,30 | +2+1 | 0.00030 |
| 0,25,31 | +3+1 | 0.00062 |
| Steps | Delta signature | Least-squares error |
|---|---|---|
| 0,1,2,3 | +1+?+1 | 0.00056 |
| 0,1,2,4 | +1+?+2 | 0.00100 |
| 0,1,3,4 | +1+?+1 | 0.00085 |
| 0,1,16,17 | +2+?+3 | 0.00091 |
| 0,1,16,18 | +1+?+3 | 0.00093 |
| 0,1,17,18 | +2+?+3 | 0.00058 |
| 0,1,17,19 | +1+?+3 | 0.00051 |
| 0,1,18,19 | +2+?+3 | 0.00025 |
| 0,1,18,20 | +1+?+3 | 0.00009 |
| 0,1,19,20 | +2+?+3 | 0.00010 |
| 0,1,19,21 | +1+?+3 | 0.00034 |
| 0,1,20,21 | +2+?+3 | 0.00045 |
| 0,1,20,22 | +1+?+3 | 0.00078 |
| 0,1,21,22 | +2+?+3 | 0.00081 |
| 0,1,30,31 | +1+?+2 | 0.00076 |
| 0,2,3,4 | +2+?+1 | 0.00082 |
| 0,2,6,11 | +1+?+3 | 0.00077 |
| 0,2,7,12 | +1+?+3 | 0.00009 |
| 0,2,8,13 | +1+?+3 | 0.00097 |
| 0,2,12,13 | +3+?+2 | 0.00072 |
| 0,2,12,15 | +1+?+2 | 0.00060 |
| 0,2,13,14 | +3+?+2 | 0.00032 |
| 0,2,13,16 | +1+?+2 | 0.00018 |
| 0,2,14,15 | +3+?+2 | 0.00009 |
| 0,2,14,17 | +1+?+2 | 0.00097 |
| 0,2,15,16 | +3+?+2 | 0.00050 |
| 0,2,16,17 | +3+?+2 | 0.00093 |
| 0,2,17,21 | +1+?+3 | 0.00061 |
| 0,2,18,20 | +2+?+3 | 0.00050 |
| 0,2,18,22 | +1+?+3 | 0.00025 |
| 0,2,19,21 | +2+?+3 | 0.00020 |
| 0,2,20,22 | +2+?+3 | 0.00091 |
| 0,3,4,8 | +2+?+3 | 0.00098 |
| 0,3,5,9 | +2+?+3 | 0.00007 |
| 0,3,7,12 | +1+?+2 | 0.00048 |
| 0,3,8,13 | +1+?+2 | 0.00071 |
| 0,3,9,16 | +1+?+3 | 0.00074 |
| 0,3,10,17 | +1+?+3 | 0.00057 |
| 0,3,17,23 | +1+?+3 | 0.00026 |
| 0,3,18,19 | +2+?+1 | 0.00082 |
| 0,3,18,21 | +2+?+3 | 0.00075 |
| 0,3,18,22 | +1+?+2 | 0.00025 |
| 0,3,19,20 | +2+?+1 | 0.00035 |
| 0,3,19,21 | +1+?+1 | 0.00019 |
| 0,3,19,22 | +2+?+3 | 0.00030 |
| 0,3,19,23 | +1+?+2 | 0.00094 |
| 0,3,20,21 | +2+?+1 | 0.00013 |
| 0,3,20,22 | +1+?+1 | 0.00066 |
| 0,3,21,22 | +2+?+1 | 0.00063 |
| 0,3,26,31 | +1+?+3 | 0.00016 |
| 0,4,5,12 | +1+?+2 | 0.00059 |
| 0,4,5,15 | +1+?+3 | 0.00060 |
| 0,4,8,13 | +2+?+3 | 0.00013 |
| 0,4,11,20 | +1+?+3 | 0.00049 |
| 0,4,12,18 | +1+?+2 | 0.00042 |
| 0,4,13,14 | +3+?+1 | 0.00079 |
| 0,4,13,16 | +1+?+1 | 0.00088 |
| 0,4,14,15 | +3+?+1 | 0.00035 |
| 0,4,14,16 | +3+?+2 | 0.00024 |
| 0,4,14,17 | +1+?+1 | 0.00024 |
| 0,4,15,16 | +3+?+1 | 0.00009 |
| 0,4,15,17 | +3+?+2 | 0.00060 |
| 0,4,16,17 | +3+?+1 | 0.00055 |
| 0,4,17,25 | +1+?+3 | 0.00058 |
| 0,4,19,23 | +2+?+3 | 0.00040 |
| 0,4,21,26 | +1+?+2 | 0.00030 |
| 0,4,23,30 | +1+?+3 | 0.00062 |
| 0,5,6,9 | +3+?+2 | 0.00013 |
| 0,5,7,19 | +1+?+3 | 0.00069 |
| 0,5,9,17 | +1+?+2 | 0.00047 |
| 0,5,10,16 | +2+?+3 | 0.00038 |
| 0,5,11,13 | +2+?+1 | 0.00067 |
| 0,5,11,15 | +1+?+1 | 0.00027 |
| 0,5,11,22 | +1+?+3 | 0.00052 |
| 0,5,12,14 | +2+?+1 | 0.00015 |
| 0,5,13,15 | +2+?+1 | 0.00099 |
| 0,5,15,22 | +1+?+2 | 0.00090 |
| 0,5,16,26 | +1+?+3 | 0.00034 |
| 0,5,19,24 | +2+?+3 | 0.00051 |
| 0,5,23,29 | +1+?+2 | 0.00015 |
| 0,5,24,25 | +3+?+1 | 0.00090 |
| 0,5,24,27 | +1+?+1 | 0.00085 |
| 0,5,25,26 | +3+?+1 | 0.00034 |
| 0,5,25,27 | +3+?+2 | 0.00011 |
| 0,5,25,28 | +1+?+1 | 0.00058 |
| 0,5,26,27 | +3+?+1 | 0.00023 |
| 0,5,26,28 | +3+?+2 | 0.00096 |
| 0,5,27,28 | +3+?+1 | 0.00081 |
| 0,6,9,14 | +1+?+1 | 0.00013 |
| 0,6,11,18 | +2+?+3 | 0.00020 |
| 0,6,12,21 | +1+?+2 | 0.00064 |
| 0,6,15,18 | +3+?+2 | 0.00025 |
| 0,6,18,26 | +1+?+2 | 0.00075 |
| 0,6,19,25 | +2+?+3 | 0.00062 |
| 0,6,20,22 | +2+?+1 | 0.00074 |
| 0,6,20,24 | +1+?+1 | 0.00046 |
| 0,6,20,31 | +1+?+3 | 0.00043 |
| 0,6,21,23 | +2+?+1 | 0.00025 |
| 0,6,24,31 | +1+?+2 | 0.00091 |
| 0,7,8,12 | +3+?+2 | 0.00097 |
| 0,7,8,14 | +1+?+1 | 0.00076 |
| 0,7,8,24 | +1+?+3 | 0.00043 |
| 0,7,9,11 | +3+?+1 | 0.00053 |
| 0,7,9,12 | +2+?+1 | 0.00018 |
| 0,7,9,13 | +3+?+2 | 0.00054 |
| 0,7,9,20 | +1+?+2 | 0.00020 |
| 0,7,10,12 | +3+?+1 | 0.00028 |
| 0,7,12,20 | +2+?+3 | 0.00010 |
| 0,7,14,24 | +1+?+2 | 0.00004 |
| 0,7,15,29 | +1+?+3 | 0.00028 |
| 0,7,17,22 | +1+?+1 | 0.00091 |
| 0,7,19,26 | +2+?+3 | 0.00073 |
| 0,7,22,25 | +3+?+2 | 0.00065 |
| 0,7,23,26 | +3+?+2 | 0.00086 |
| 0,7,27,31 | +1+?+1 | 0.00074 |
| 0,7,28,30 | +2+?+1 | 0.00044 |
| 0,7,29,31 | +2+?+1 | 0.00074 |
| 0,8,11,23 | +1+?+2 | 0.00070 |
| 0,8,11,28 | +1+?+3 | 0.00080 |
| 0,8,13,22 | +2+?+3 | 0.00070 |
| 0,8,14,20 | +1+?+1 | 0.00072 |
| 0,8,15,19 | +3+?+2 | 0.00057 |
| 0,8,16,18 | +3+?+1 | 0.00031 |
| 0,8,16,19 | +2+?+1 | 0.00023 |
| 0,8,16,27 | +1+?+2 | 0.00085 |
| 0,8,17,19 | +3+?+1 | 0.00063 |
| 0,8,19,27 | +2+?+3 | 0.00084 |
| 0,8,23,28 | +1+?+1 | 0.00055 |
| 0,9,10,15 | +3+?+2 | 0.00092 |
| 0,9,11,30 | +1+?+3 | 0.00012 |
| 0,9,13,20 | +1+?+1 | 0.00100 |
| 0,9,13,26 | +1+?+2 | 0.00021 |
| 0,9,17,29 | +1+?+2 | 0.00062 |
| 0,9,19,28 | +2+?+3 | 0.00096 |
| 0,9,20,26 | +1+?+1 | 0.00070 |
| 0,9,21,25 | +3+?+2 | 0.00055 |
| 0,9,22,24 | +3+?+1 | 0.00031 |
| 0,9,22,25 | +2+?+1 | 0.00034 |
| 0,9,23,25 | +3+?+1 | 0.00077 |
| 0,10,13,17 | +2+?+1 | 0.00066 |
| 0,10,14,25 | +2+?+3 | 0.00076 |
| 0,10,16,21 | +3+?+2 | 0.00034 |
| 0,10,18,25 | +1+?+1 | 0.00004 |
| 0,10,27,29 | +3+?+1 | 0.00080 |
| 0,10,27,30 | +2+?+1 | 0.00029 |
| 0,10,27,31 | +3+?+2 | 0.00077 |
| 0,10,28,30 | +3+?+1 | 0.00040 |
| 0,11,12,18 | +3+?+2 | 0.00040 |
| 0,11,12,28 | +1+?+2 | 0.00038 |
| 0,11,13,16 | +3+?+1 | 0.00049 |
| 0,11,14,17 | +3+?+1 | 0.00085 |
| 0,11,14,26 | +2+?+3 | 0.00077 |
| 0,11,16,24 | +1+?+1 | 0.00085 |
| 0,11,18,22 | +2+?+1 | 0.00057 |
| 0,11,21,26 | +3+?+2 | 0.00058 |
| 0,11,23,30 | +1+?+1 | 0.00023 |
| 0,12,15,24 | +1+?+1 | 0.00060 |
| 0,12,18,21 | +3+?+1 | 0.00014 |
| 0,12,21,29 | +1+?+1 | 0.00078 |
| 0,12,23,27 | +2+?+1 | 0.00036 |
| 0,12,25,30 | +3+?+2 | 0.00084 |
| 0,13,16,21 | +2+?+1 | 0.00057 |
| 0,13,19,28 | +1+?+1 | 0.00023 |
| 0,13,22,25 | +3+?+1 | 0.00019 |
| 0,13,27,31 | +2+?+1 | 0.00012 |
| 0,14,15,30 | +2+?+3 | 0.00004 |
| 0,14,17,24 | +3+?+2 | 0.00028 |
| 0,14,20,25 | +2+?+1 | 0.00048 |
| 0,14,26,29 | +3+?+1 | 0.00012 |
| 0,15,16,20 | +3+?+1 | 0.00002 |
| 0,15,24,29 | +2+?+1 | 0.00028 |
| 0,16,20,31 | +1+?+1 | 0.00042 |
| 0,16,24,31 | +3+?+2 | 0.00051 |
| 0,17,21,29 | +3+?+2 | 0.00090 |
| 0,17,22,28 | +2+?+1 | 0.00062 |
| 0,17,23,27 | +3+?+1 | 0.00039 |
| 0,18,25,31 | +2+?+1 | 0.00007 |
| 0,18,26,30 | +3+?+1 | 0.00001 |
| 0,19,21,30 | +3+?+2 | 0.00014 |
| 0,20,21,26 | +3+?+1 | 0.00032 |
| 0,21,24,29 | +3+?+1 | 0.00026 |
Octave stretch or compression
Whether octave stretch, shrink or neither is advised for 32edo depends on which vals one wishes to use.
For 32, pure-octaves, or slight compression (~0.5 ¢), works well.
For 32f, moderate compression (~2 ¢) works well. This is close to 133zpi (32.07edo).
For 32c or 32cf, substantial compression (3-4 ¢) is well suited.
For 32be, substantial stretch works (~5 ¢). This is close to 132zpi (31.86edo).
The graph shows zeta near 32edo.
Scales
A list of useful scales known so far in 32edo.
Named
- Antipental blues: 9 5 1 4 9 4
Unnamed
- Bluesy spy music-type scale no. 1: 7 3 9 9 4
- Bluesy spy music-type scale no. 2: 9 6 1 3 13
- Nocturnal fay-type scale: 5 14 3 3 7
- Pseudo-kokinjoshi no. 1: 6 2 11 6 7
- Pseudo-kokinjoshi no. 2: 3 10 5 9 5
- Pseudo-kokinjoshi no. 3: 5 9 5 3 10
- Pseudo-pelog #m3 #m6: 3 5 10 4 10
- Pseudo-pelog ##m3 #P5 #m6: 3 6 10 3 10
- Robotic-type scale, good for contemporary classical: 16 3 4 3 6
- Sunny crystalline-type scale no. 1: 4 4 6 5 8 5
- Sunny crystalline-type scale no. 2: 9 6 4 4 4 5
Instruments
Music
Modern renderings
- "Lost Woods" from The Legend of Zelda: Ocarina of Time OST (1998) – covered by Bryan Deister (2025)
21st century
- 32edo improv (2025)
- Licorice Hearted (2026)
- "Winter's Mortal Hope", from A New Dusk (2024) – Bandcamp | YouTube (22:37–26:00)
- Canon on Twinkle Twinkle Little Star
- for organ (2023) · for baroque oboe & viola (2024)
- 1332 (2019)
- 8 vs 16 vs 32 Notes Per Octave — The 96-Tone Piano in Its Subdivision Glory (2026, performed on a digital recreation of Julián Carrillo's 96-tone metamorphoser piano, using subsets 8edo, 16edo, 32edo, and then glissandos in full 96edo)
- "Zinnia Riplet", featured in Possible Worlds Vol. 4 (2019) of Spectropol Records
- Admin's Hot Tub (2019)
- Beyond the Grid (2024)
- Slither (2025)
- Cipher (2025)
- Revelation (2026)
