32ed12

← 31ed12

32ed12

33ed12 →

Prime factorization	25
Step size	134.436¢
Octave	9\32ed12 (1209.92¢)
Twelfth	14\32ed12 (1882.11¢) (→7\16ed12)
Consistency limit	8
Distinct consistency limit	6

32 equal divisions of the 12th harmonic (abbreviated 32ed12) is a nonoctave tuning system that divides the interval of 12/1 into 32 equal parts of about 134⁠ ⁠¢ each. Each step represents a frequency ratio of 12^1/32, or the 32nd root of 12.

32ed12 is similar to 9edo, but has the 12/1 tuned just instead of the octave, which stretches the octave by 9.9 ¢. It also approximates Pelog tunings in Indonesian gamelan music very well, since Pelog is well-approximated by 9edo with stretched octaves.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	134.4	13/12, 14/13
2	268.9	7/6, 27/23
3	403.3	19/15, 24/19
4	537.7	15/11, 19/14, 26/19
5	672.2	22/15, 28/19
6	806.6	8/5, 27/17
7	941.1	12/7, 19/11
8	1075.5	13/7, 28/15
9	1209.9
10	1344.4	13/6, 24/11
11	1478.8
12	1613.2	28/11
13	1747.7	11/4
14	1882.1
15	2016.5	16/5
16	2151
17	2285.4	15/4
18	2419.8
19	2554.3
20	2688.7	19/4
21	2823.2
22	2957.6	11/2
23	3092
24	3226.5
25	3360.9	7/1
26	3495.3	15/2
27	3629.8
28	3764.2
29	3898.6	19/2
30	4033.1
31	4167.5
32	4302	12/1

Harmonics

Approximation of harmonics in 32ed12

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+9.9	-19.8	+19.8	+36.8	-9.9	-7.9	+29.8	-39.7	+46.8	+16.2	+0.0
	Relative (%)	+7.4	-14.8	+14.8	+27.4	-7.4	-5.9	+22.1	-29.5	+34.8	+12.1	+0.0
Steps (reduced)		9 (9)	14 (14)	18 (18)	21 (21)	23 (23)	25 (25)	27 (27)	28 (28)	30 (30)	31 (31)	32 (0)

Approximation of harmonics in 32ed12

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-4.1	+2.0	+17.0	+39.7	-65.3	-29.8	+11.1	+56.7	-27.8	+26.1	-50.8
	Relative (%)	-3.1	+1.5	+12.6	+29.5	-48.5	-22.1	+8.2	+42.2	-20.7	+19.4	-37.8
Steps (reduced)		33 (1)	34 (2)	35 (3)	36 (4)	36 (4)	37 (5)	38 (6)	39 (7)	39 (7)	40 (8)	40 (8)