32edo

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.

Music

Sixix by Petr Parízek

32 32 32 Nothing Less Will Do by Chris Vaisvil