57ed4

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57ed4

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Prime factorization	3 × 19
Step size	42.1053 ¢
Octave	29\57ed4 (1221.05 ¢)
Twelfth	45\57ed4 (1894.74 ¢) (→ 15\19ed4)
Consistency limit	1
Distinct consistency limit	1

57 equal divisions of the 4th harmonic (abbreviated 57ed4) is a nonoctave tuning system that divides the interval of 4/1 into 57 equal parts of about 42.1 ¢ each. Each step represents a frequency ratio of 4^1/57, or the 57th root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	42.1
2	84.2
3	126.3	29/27
4	168.4	11/10
5	210.5	26/23
6	252.6	29/25
7	294.7
8	336.8
9	378.9
10	421.1	14/11, 23/18
11	463.2	17/13, 30/23
12	505.3
13	547.4	26/19
14	589.5
15	631.6
16	673.7	31/21, 34/23
17	715.8
18	757.9
19	800	27/17
20	842.1
21	884.2	5/3
22	926.3	29/17
23	968.4
24	1010.5	34/19
25	1052.6	11/6
26	1094.7
27	1136.8	25/13, 29/15
28	1178.9
29	1221.1
30	1263.2	27/13
31	1305.3
32	1347.4
33	1389.5	29/13
34	1431.6
35	1473.7
36	1515.8
37	1557.9
38	1600
39	1642.1
40	1684.2
41	1726.3	19/7
42	1768.4	25/9
43	1810.5
44	1852.6
45	1894.7
46	1936.8
47	1978.9
48	2021.1	29/9
49	2063.2	23/7, 33/10
50	2105.3
51	2147.4
52	2189.5
53	2231.6
54	2273.7	26/7
55	2315.8	19/5
56	2357.9
57	2400

Harmonics

Approximation of harmonics in 57ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+21.1	-7.2	+0.0	-7.4	+13.8	-0.4	+21.1	-14.4	+13.7	+17.1	-7.2
	Relative (%)	+50.0	-17.1	+0.0	-17.5	+32.9	-1.0	+50.0	-34.3	+32.5	+40.6	-17.1
Steps (reduced)		29 (29)	45 (45)	57 (0)	66 (9)	74 (17)	80 (23)	86 (29)	90 (33)	95 (38)	99 (42)	102 (45)

Approximation of harmonics in 57ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-19.5	+20.6	-14.6	+0.0	-20.7	+6.6	-2.8	-7.4	-7.6	-3.9	+3.3
	Relative (%)	-46.3	+49.0	-34.6	+0.0	-49.3	+15.7	-6.6	-17.5	-18.1	-9.4	+7.8
Steps (reduced)		105 (48)	109 (52)	111 (54)	114 (0)	116 (2)	119 (5)	121 (7)	123 (9)	125 (11)	127 (13)	129 (15)