57ed5

← 56ed5

57ed5

58ed5 →

Prime factorization	3 × 19
Step size	48.8827 ¢
Octave	25\57ed5 (1222.07 ¢)
Twelfth	39\57ed5 (1906.43 ¢) (→ 13\19ed5)
Consistency limit	3
Distinct consistency limit	3

57 equal divisions of the 5th harmonic (abbreviated 57ed5) is a nonoctave tuning system that divides the interval of 5/1 into 57 equal parts of about 48.9 ¢ each. Each step represents a frequency ratio of 5^1/57, or the 57th root of 5.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	48.9	34/33
2	97.8
3	146.6	25/23
4	195.5	19/17
5	244.4	15/13
6	293.3	13/11
7	342.2
8	391.1
9	439.9
10	488.8
11	537.7	15/11
12	586.6	7/5
13	635.5	13/9
14	684.4
15	733.2	29/19
16	782.1	11/7
17	831	21/13, 34/21
18	879.9	5/3
19	928.8	29/17
20	977.7
21	1026.5
22	1075.4	13/7
23	1124.3
24	1173.2
25	1222.1
26	1271
27	1319.8	15/7
28	1368.7	11/5
29	1417.6	25/11, 34/15
30	1466.5	7/3
31	1515.4
32	1564.2
33	1613.1	33/13
34	1662	34/13
35	1710.9
36	1759.8
37	1808.7
38	1857.5
39	1906.4	3/1
40	1955.3	31/10, 34/11
41	2004.2
42	2053.1
43	2102
44	2150.8
45	2199.7	25/7
46	2248.6	11/3
47	2297.5	34/9
48	2346.4
49	2395.3
50	2444.1
51	2493
52	2541.9	13/3
53	2590.8
54	2639.7	23/5
55	2688.5	33/7
56	2737.4	34/7
57	2786.3	5/1

Harmonics

Approximation of harmonics in 57ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+22.1	+4.5	-4.7	+0.0	-22.3	+4.1	+17.3	+8.9	+22.1	+3.7	-0.3
	Relative (%)	+45.1	+9.1	-9.7	+0.0	-45.7	+8.3	+35.4	+18.3	+45.1	+7.6	-0.6
Steps (reduced)		25 (25)	39 (39)	49 (49)	57 (0)	63 (6)	69 (12)	74 (17)	78 (21)	82 (25)	85 (28)	88 (31)

Approximation of harmonics in 57ed5 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+7.8	-22.7	+4.5	-9.5	-16.7	-17.9	-13.7	-4.7	+8.6	-23.1	-2.3
	Relative (%)	+16.0	-46.5	+9.1	-19.4	-34.1	-36.6	-28.1	-9.7	+17.5	-47.3	-4.7
Steps (reduced)		91 (34)	93 (36)	96 (39)	98 (41)	100 (43)	102 (45)	104 (47)	106 (49)	108 (51)	109 (52)	111 (54)

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