32edo

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The 32 equal division divides the octave into 32 equal parts of precisely 37.5 cents each. While even advocates of less-common EDOs can struggle to find something about it worth noting, it does provide an excellent tuning for Petr Parízek's sixix temperament, which tempers out the 5-limit sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents. Parízek'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 temperament, 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.

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 v~2 downmid 2nd ^^Eb 16/15 49/46
4 150.0 ^~2 upmid 2nd vvE 12/11, 49/45 25/23
5 187.5 vM2 downmajor 2nd 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 v~3 downmid 3rd ^^F 11/9, 39/32, 63/52 17/14, 28/23
10 375.0 ^~3 upmid 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 v~4 downmid 4th ^^G 18/13, 11/8
16 600.0 ^~4,v~5 upmid 4th, downmid 5th vvG#, ^^Ab 7/5, 10/7, 99/70, 140/99 17/12, 12/17
17 637.5 ^~5 upmid 5th vvA 13/9, 16/11
18 675.0 v5 down fifth 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 v~6 downmid 6th ^^Bb 8/5, 21/13, 45/28, 77/48 29/18
23 862.5 ^~6 upmid 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 v~7 downmid 7th ^^C 11/6, 90/49 46/25
29 1087.5 ^~7 upmid 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 major 7th C# 96/49, 49/25, 88/45 45/23, 51/26, 100/51
32 1200.0 P8 8ve D 2/1

Z 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.

plot32.png

Music

Zinnia Riplet by Stephen Weigel (featured in Possible Worlds Vol. 4 of Spectropol Records)

Admin's Hot Tub by Stephen Weigel

Sixix by Petr Parízek

32 32 32 Nothing Less Will Do by Chris Vaisvil