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.

Harmonics

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.

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

Notation

Sagittal notation

This notation uses the same sagittal sequence as 25-EDO, and is a subset of the notation for 64b.

Evo flavor

Revo flavor

Ups and downs notation

Another way to notate the microtonal notes is to use ups and downs. Here, this can be done using sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:

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 double arrows.

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.

15-odd-limit intervals in 32edo (direct approximation, even if inconsistent)
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

15-odd-limit intervals in 32edo (patent val mapping)
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

Zeta properties

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.

Zeta peak index

Tuning			Strength			Closest edo		Integer limit
ZPI	Steps per octave	Step size (cents)	Height	Integral	Gap	Edo	Octave (cents)	Consistent	Distinct
133zpi	32.0701500181780	37.4179727665700	4.471728	0.689412	12.537826	32edo	1197.37512853024	4	4

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