53ed4

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53ed4

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Prime factorization	53 (prime)
Step size	45.283¢
Octave	27\53ed4 (1222.64¢)
Twelfth	42\53ed4 (1901.89¢) (convergent)
Consistency limit	1
Distinct consistency limit	1

53 equal divisions of the 4th harmonic (abbreviated 53ed4) is a nonoctave tuning system that divides the interval of 4/1 into 53 equal parts of about 45.3 ¢ each. Each step represents a frequency ratio of 4^1/53, or the 53rd root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	45.3
2	90.6	19/18
3	135.8
4	181.1
5	226.4	25/22, 33/29
6	271.7
7	317	6/5
8	362.3	21/17
9	407.5	19/15, 33/26
10	452.8
11	498.1
12	543.4	26/19
13	588.7
14	634	13/9
15	679.2	31/21
16	724.5
17	769.8	14/9
18	815.1
19	860.4	23/14
20	905.7
21	950.9	19/11, 26/15, 33/19
22	996.2
23	1041.5	31/17
24	1086.8
25	1132.1	27/14
26	1177.4
27	1222.6
28	1267.9	25/12, 27/13
29	1313.2
30	1358.5
31	1403.8
32	1449.1
33	1494.3	26/11
34	1539.6	17/7
35	1584.9	5/2
36	1630.2
37	1675.5	29/11
38	1720.8
39	1766
40	1811.3
41	1856.6
42	1901.9	3/1
43	1947.2
44	1992.5	19/6
45	2037.7
46	2083
47	2128.3
48	2173.6
49	2218.9	18/5
50	2264.2
51	2309.4	19/5
52	2354.7
53	2400

Harmonics

Approximation of harmonics in 53ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+22.6	-0.1	+0.0	+21.2	+22.6	-17.9	+22.6	-0.1	-1.4	+14.7	-0.1
	Relative (%)	+50.0	-0.2	+0.0	+46.9	+49.8	-39.5	+50.0	-0.3	-3.1	+32.5	-0.2
Steps (reduced)		27 (27)	42 (42)	53 (0)	62 (9)	69 (16)	74 (21)	80 (27)	84 (31)	88 (35)	92 (39)	95 (42)

Approximation of harmonics in 53ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.8	+4.8	+21.2	+0.0	-14.4	+22.5	+19.5	+21.2	-18.0	-7.9	+5.7
	Relative (%)	-6.2	+10.5	+46.7	+0.0	-31.8	+49.7	+43.0	+46.9	-39.6	-17.5	+12.6
Steps (reduced)		98 (45)	101 (48)	104 (51)	106 (0)	108 (2)	111 (5)	113 (7)	115 (9)	116 (10)	118 (12)	120 (14)