20567edo

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20567edo

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Prime factorization	131 × 157
Step size	0.0583459¢
Fifth	12031\20567 (701.959¢)
Semitones (A1:m2)	1949:1546 (113.7¢ : 90.2¢)
Consistency limit	57
Distinct consistency limit	57

20567 equal divisions of the octave (abbreviated 20567edo or 20567ed2), also called 20567-tone equal temperament (20567tet) or 20567 equal temperament (20567et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 20567 equal parts of about 0.0583 ¢ each. Each step represents a frequency ratio of 2^1/20567, or the 20567th root of 2.

20567edo is a remarkable very high-limit system, distinctly consistent through the 57-odd-limit, with a lower relative error than any previous equal temperaments in the 43-limit. It tempers out 33814/33813, 35344/35343, 37180/37179, 42484/42483, 42688/42687, 47125/47124, 48504/48503, 67915/67914, 70500/70499, 91885/91884, 126225/126224, 156520/156519, 194580/194579, 206800/206793, and 561925/561924 in the 53-limit.

Prime harmonics

Approximation of prime harmonics in 20567edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	+0.0000	+0.0044	-0.0056	+0.0077	-0.0076	+0.0033	+0.0089	-0.0073	-0.0058	-0.0056	+0.0026	+0.0101
	Relative (%)	+0.0	+7.6	-9.5	+13.1	-13.0	+5.6	+15.2	-12.5	-9.9	-9.5	+4.4	+17.3
Steps (reduced)		20567 (0)	32598 (12031)	47755 (6621)	57739 (16605)	71150 (9449)	76107 (14406)	84067 (1799)	87367 (5099)	93036 (10768)	99914 (17646)	101893 (19625)	107143 (4308)

Approximation of prime harmonics in 20567edo (continued)

Harmonic		41	43	47	53	59	61	67	71	73	79	83	89
Error	Absolute (¢)	+0.0133	+0.0007	-0.0134	-0.0082	-0.0186	-0.0277	-0.0150	+0.0088	-0.0071	+0.0082	-0.0254	-0.0239
	Relative (%)	+22.8	+1.3	-22.9	-14.0	-32.0	-47.5	-25.6	+15.1	-12.2	+14.1	-43.6	-40.9
Steps (reduced)		110189 (7354)	111602 (8767)	114241 (11406)	117806 (14971)	120988 (18153)	121977 (19142)	124761 (1359)	126482 (3080)	127306 (3904)	129650 (6248)	131115 (7713)	133186 (9784)