32edo

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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 2^1/32, 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 9/7 and 13/10, 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 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.

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
Error	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

Approximation to JI

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.

Delta-rational harmony

The tables below show chords that approximate 3-integer-limit delta-rational chords with least-squares error less than 0.0015.

Fully delta-rational triads

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

Partially delta-rational tetrads

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