# 32edo

 ← 31edo 32edo 33edo →
Prime factorization 25
Step size 37.5¢
Fifth 19\32 (712.5¢)
Semitones (A1:m2) 5:1 (187.5¢ : 37.5¢)
Consistency limit 3
Distinct consistency limit 3

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

While even advocates of less-common edos can struggle to find something about 32edo worth noting, it does provide an excellent 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. 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. It also can be used (with the 9\32 generator) to tune mohavila, an 11-limit temperament which does not temper out sixix.

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 the 15 & 32 temperament, tempering out 55/54, 64/63 and 245/242.

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 "major third" (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating 13/10~9/7, and the minor third is 262.5¢, approximating 7/6. Because of the unequalness of the scale, the minor second is reduced to a fifth-tone, but it still strongly resembles "normal" diatonic music, especially for darker modes. 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 and discordant than the sharp fifth.

It is generally the first power-of-2 edo which can be considered to handle low-limit JI 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 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.

### Harmonics

Approximation of odd harmonics in 32edo
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)

## 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

## JI approximation

### Zeta function

Below is a plot of the Zeta function, showing how its peak (ie most negative) value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.

Claudi Meneghin
Petr Pařízek
Chris Vaisvil
Stephen Weigel