57ed6

← 56ed6

57ed6

58ed6 →

Prime factorization	3 × 19
Step size	54.4203¢
Octave	22\57ed6 (1197.25¢)
Twelfth	35\57ed6 (1904.71¢)
Consistency limit	12
Distinct consistency limit	8

57 equal divisions of the 6th harmonic (abbreviated 57ed6) is a nonoctave tuning system that divides the interval of 6/1 into 57 equal parts of about 54.4⁠ ⁠¢ each. Each step represents a frequency ratio of 6^1/57, or the 57th root of 6.

57ed6 is very nearly identical to 22 EDO, but with the 6/1 rather than the 2/1 being just, which results in octaves being compressed by about 2.754 ¢. The local zeta peak around 22 is located at 22.025147, which has a step size of 54.483 ¢ and an octave of 1198.63 ¢ (which is compressed by 1.37 ¢), making 57ed6 very close to optimal for 22edo.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	54.4	30/29, 31/30, 32/31, 33/32, 34/33
2	108.8	16/15, 17/16, 33/31
3	163.3	11/10, 34/31
4	217.7	17/15, 25/22
5	272.1	34/29
6	326.5	23/19, 29/24
7	380.9
8	435.4	9/7
9	489.8
10	544.2	26/19
11	598.6	17/12, 24/17
12	653	16/11, 19/13
13	707.5
14	761.9	14/9, 31/20
15	816.3	8/5
16	870.7	33/20
17	925.1	29/17
18	979.6	30/17
19	1034	20/11, 29/16
20	1088.4	15/8
21	1142.8	29/15, 31/16
22	1197.2	2/1
23	1251.7	33/16
24	1306.1	17/8
25	1360.5	11/5
26	1414.9	34/15
27	1469.3	7/3
28	1523.8	29/12
29	1578.2
30	1632.6	18/7
31	1687
32	1741.4	30/11
33	1795.9	31/11
34	1850.3	32/11
35	1904.7	3/1
36	1959.1	31/10
37	2013.5	16/5
38	2068	33/10
39	2122.4	17/5
40	2176.8
41	2231.2	29/8
42	2285.7	15/4
43	2340.1	27/7
44	2394.5
45	2448.9	33/8
46	2503.3	17/4
47	2557.8
48	2612.2
49	2666.6	14/3
50	2721
51	2775.4
52	2829.9
53	2884.3
54	2938.7
55	2993.1
56	3047.5	29/5
57	3102	6/1

Harmonics

Approximation of harmonics in 57ed6

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-2.8	+2.8	-5.5	-10.9	+0.0	+5.2	-8.3	+5.5	-13.6	-15.4	-2.8
	Relative (%)	-5.1	+5.1	-10.1	-20.0	+0.0	+9.6	-15.2	+10.1	-25.1	-28.3	-5.1
Steps (reduced)		22 (22)	35 (35)	44 (44)	51 (51)	57 (0)	62 (5)	66 (9)	70 (13)	73 (16)	76 (19)	79 (22)

Approximation of harmonics in 57ed6

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+21.9	+2.5	-8.1	-11.0	-7.1	+2.8	+18.0	-16.4	+8.0	-18.1	+13.8
	Relative (%)	+40.3	+4.6	-14.9	-20.2	-13.1	+5.1	+33.1	-30.1	+14.7	-33.3	+25.3
Steps (reduced)		82 (25)	84 (27)	86 (29)	88 (31)	90 (33)	92 (35)	94 (37)	95 (38)	97 (40)	98 (41)	100 (43)

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