57ed12

57ED12 is the equal division of the twelfth harmonic into 57 equal parts. It is similar to 16edo, but has the 12/1 tuned just instead of the octave. Octaves are stretched by 7.6 ¢ and the step size is 75.47 ¢. It can be used as a tuning for Mavila and also approximates Pelog tunings in Indonesian gamelan music.

← 56ed12

57ed12

58ed12 →

Prime factorization	3 × 19
Step size	75.4729 ¢
Octave	16\57ed12 (1207.57 ¢)
Twelfth	25\57ed12 (1886.82 ¢)
Consistency limit	6
Distinct consistency limit	6

Theory

This tuning tempers out 36/35 and 50/49 in the 7-limit; 33/32 and 45/44 in the 11-limit; 65/64, 66/65, and 78/77 in the 13-limit; 51/50 and 85/84 in the 17-limit; 39/38, 57/56, 77/76, and 96/95 in the 19-limit; 69/68 and 92/91 in the 23-limit; 58/57 and 87/85 in the 29-limit; 63/62, 93/92, and 93/91 in the 31-limit; and 75/74 in the 37-limit.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	75.5	23/22, 24/23
2	150.9	12/11
3	226.4	8/7, 33/29
4	301.9	19/16, 25/21, 31/26
5	377.4
6	452.8	13/10
7	528.3	19/14, 34/25
8	603.8	17/12
9	679.3	34/23
10	754.7	17/11, 31/20
11	830.2	21/13, 29/18, 34/21
12	905.7	32/19
13	981.1	30/17
14	1056.6
15	1132.1	25/13
16	1207.6
17	1283	21/10
18	1358.5
19	1434	16/7
20	1509.5
21	1584.9	5/2
22	1660.4	34/13
23	1735.9	30/11
24	1811.3
25	1886.8
26	1962.3	31/10
27	2037.8	13/4
28	2113.2
29	2188.7
30	2264.2
31	2339.7
32	2415.1
33	2490.6
34	2566.1	22/5
35	2641.6	23/5
36	2717	24/5
37	2792.5
38	2868	21/4
39	2943.4
40	3018.9
41	3094.4
42	3169.9	25/4
43	3245.3
44	3320.8	34/5
45	3396.3
46	3471.8
47	3547.2	31/4
48	3622.7
49	3698.2
50	3773.6
51	3849.1
52	3924.6	29/3
53	4000.1
54	4075.5	21/2
55	4151	11/1
56	4226.5	23/2
57	4302	12/1

Harmonics

Approximation of harmonics in 57ed12

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+7.6	-15.1	+15.1	+6.2	-7.6	+27.5	+22.7	-30.3	+13.7	-0.3	+0.0
	Relative (%)	+10.0	-20.1	+20.1	+8.2	-10.0	+36.4	+30.1	-40.1	+18.2	-0.4	+0.0
Steps (reduced)		16 (16)	25 (25)	32 (32)	37 (37)	41 (41)	45 (45)	48 (48)	50 (50)	53 (53)	55 (55)	57 (0)

Approximation of harmonics in 57ed12

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+12.4	+35.0	-8.9	+30.3	+0.8	-22.7	+34.6	+21.3	+12.3	+7.3	+5.8
	Relative (%)	+16.4	+46.4	-11.9	+40.1	+1.0	-30.1	+45.9	+28.2	+16.3	+9.6	+7.7
Steps (reduced)		59 (2)	61 (4)	62 (5)	64 (7)	65 (8)	66 (9)	68 (11)	69 (12)	70 (13)	71 (14)	72 (15)