20567edo

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20567edo

20568edo →

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.

Theory

20567edo is a remarkable very high-limit system, distinctly (and almost purely, as all odd harmonics 57 and below, except 49, are within 25% relative error) 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
Error	Absolute (¢)	+0.0000	+0.0044	-0.0056	+0.0077	-0.0076	+0.0033	+0.0089	-0.0073	-0.0058	-0.0056	+0.0026
	Relative (%)	+0.0	+7.6	-9.5	+13.1	-13.0	+5.6	+15.2	-12.5	-9.9	-9.5	+4.4
Steps (reduced)		20567 (0)	32598 (12031)	47755 (6621)	57739 (16605)	71150 (9449)	76107 (14406)	84067 (1799)	87367 (5099)	93036 (10768)	99914 (17646)	101893 (19625)

Approximation of prime harmonics in 20567edo (continued)

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	+0.0101	+0.0133	+0.0007	-0.0134	-0.0082	-0.0186	-0.0277	-0.0150	+0.0088	-0.0071	+0.0082
	Relative (%)	+17.3	+22.8	+1.3	-22.9	-14.0	-32.0	-47.5	-25.6	+15.1	-12.2	+14.1
Steps (reduced)		107143 (4308)	110189 (7354)	111602 (8767)	114241 (11406)	117806 (14971)	120988 (18153)	121977 (19142)	124761 (1359)	126482 (3080)	127306 (3904)	129650 (6248)

Approximation to JI

The following table shows how 57-odd-limit intervals are represented in 20567edo. Prime harmonics are in bold.

As 20567edo is consistent in the 57-odd-limit, the mappings by direct approximation and through the patent val are identical.

57-odd-limit intervals in 20567edo

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
29/20, 40/29	0.000	0.0
51/41, 82/51	0.000	0.0
17/9, 18/17	0.000	0.1
41/27, 54/41	0.000	0.1
45/26, 52/45	0.000	0.1
39/28, 56/39	0.000	0.1
23/20, 40/23	0.000	0.4
29/23, 46/29	0.000	0.4
55/47, 94/55	0.000	0.4
19/11, 22/19	0.000	0.5
35/31, 62/35	0.000	0.8
53/44, 88/53	0.001	1.0
31/26, 52/31	0.001	1.2
43/32, 64/43	0.001	1.3
45/31, 62/45	0.001	1.3
53/38, 76/53	0.001	1.5
15/8, 16/15	0.001	1.9
39/34, 68/39	0.001	1.9
13/12, 24/13	0.001	2.0
35/26, 52/35	0.001	2.0
37/36, 72/37	0.001	2.0
41/21, 42/41	0.001	2.0
17/14, 28/17	0.001	2.0
37/34, 68/37	0.001	2.1
9/7, 14/9	0.001	2.1
43/35, 70/43	0.001	2.3
23/19, 38/23	0.002	2.6
19/10, 20/19	0.002	3.0
29/19, 38/29	0.002	3.0
23/22, 44/23	0.002	3.1
43/30, 60/43	0.002	3.2
43/31, 62/43	0.002	3.2
31/24, 48/31	0.002	3.2
49/27, 54/49	0.002	3.4
51/49, 98/51	0.002	3.5
49/41, 82/49	0.002	3.5
37/21, 42/37	0.002	3.5
11/10, 20/11	0.002	3.5
29/22, 44/29	0.002	3.5
35/32, 64/35	0.002	3.6
47/25, 50/47	0.002	3.9
35/24, 48/35	0.002	4.0
39/37, 74/39	0.002	4.0
53/46, 92/53	0.002	4.1
33/29, 58/33	0.002	4.1
33/20, 40/33	0.002	4.1
37/28, 56/37	0.002	4.1
43/26, 52/43	0.003	4.4
31/16, 32/31	0.003	4.4
45/43, 86/45	0.003	4.5
53/40, 80/53	0.003	4.5
53/29, 58/53	0.003	4.5
33/23, 46/33	0.003	4.5
57/29, 58/57	0.003	4.6
57/40, 80/57	0.003	4.6
57/32, 64/57	0.003	4.9
57/46, 92/57	0.003	5.0
53/50, 100/53	0.003	5.0
33/32, 64/33	0.003	5.4
7/6, 12/7	0.003	5.5
41/37, 74/41	0.003	5.5
51/37, 74/51	0.003	5.5
21/17, 34/21	0.003	5.6
37/27, 54/37	0.003	5.6
13/8, 16/13	0.003	5.6
45/32, 64/45	0.003	5.7
25/22, 44/25	0.004	6.0
57/43, 86/57	0.004	6.2
31/30, 60/31	0.004	6.3
43/24, 48/43	0.004	6.3
25/19, 38/25	0.004	6.5
43/33, 66/43	0.004	6.7
33/19, 38/33	0.004	7.1
45/28, 56/45	0.004	7.4
13/7, 14/13	0.004	7.5
15/13, 26/15	0.004	7.5
41/36, 72/41	0.004	7.5
17/12, 24/17	0.004	7.5
41/34, 68/41	0.004	7.6
29/15, 30/29	0.004	7.6
3/2, 4/3	0.004	7.6
27/17, 34/27	0.004	7.7
23/15, 30/23	0.005	8.0
57/44, 88/57	0.005	8.1
57/35, 70/57	0.005	8.5
55/53, 106/55	0.005	8.5
53/33, 66/53	0.005	8.6
31/28, 56/31	0.005	8.7
39/31, 62/39	0.005	8.8
53/47, 94/53	0.005	8.9
49/37, 74/49	0.005	9.0
35/33, 66/35	0.005	9.0
57/53, 106/57	0.005	9.1
25/23, 46/25	0.005	9.2
57/31, 62/57	0.005	9.3
45/34, 68/45	0.006	9.4
29/16, 32/29	0.006	9.5
5/4, 8/5	0.006	9.5
29/25, 50/29	0.006	9.5
41/39, 78/41	0.006	9.5
17/13, 26/17	0.006	9.5
13/9, 18/13	0.006	9.6
39/35, 70/39	0.006	9.6
37/24, 48/37	0.006	9.6
41/28, 56/41	0.006	9.7
51/28, 56/51	0.006	9.7
27/14, 28/27	0.006	9.7
33/31, 62/33	0.006	9.8
23/16, 32/23	0.006	9.9
47/44, 88/47	0.006	9.9
55/38, 76/55	0.006	10.0
47/38, 76/47	0.006	10.4
57/52, 104/57	0.006	10.5
19/15, 30/19	0.006	10.6
31/17, 34/31	0.006	10.7
43/29, 58/43	0.006	10.8
43/40, 80/43	0.006	10.8
31/18, 36/31	0.006	10.8
49/36, 72/49	0.006	11.0
33/26, 52/33	0.006	11.0
49/34, 68/49	0.006	11.1
15/11, 22/15	0.006	11.1
43/23, 46/43	0.007	11.2
45/37, 74/45	0.007	11.5
35/34, 68/35	0.007	11.6
35/18, 36/35	0.007	11.6
37/26, 52/37	0.007	11.6
43/28, 56/43	0.007	11.9
43/39, 78/43	0.007	12.0
53/30, 60/53	0.007	12.1
19/16, 32/19	0.007	12.5
55/46, 92/55	0.007	12.6
37/31, 62/37	0.007	12.8
49/39, 78/49	0.008	13.0
11/8, 16/11	0.008	13.0
55/29, 58/55	0.008	13.0
47/46, 92/47	0.008	13.0
35/29, 58/35	0.008	13.1
7/4, 8/7	0.008	13.1
39/32, 64/39	0.008	13.3
47/40, 80/47	0.008	13.4
47/29, 58/47	0.008	13.4
35/23, 46/35	0.008	13.5
33/25, 50/33	0.008	13.6
37/35, 70/37	0.008	13.6
43/38, 76/43	0.008	13.8
43/34, 68/43	0.008	13.9
31/29, 58/31	0.008	14.0
31/20, 40/31	0.008	14.0
43/36, 72/43	0.008	14.0
53/32, 64/53	0.008	14.0
57/50, 100/57	0.008	14.1
43/22, 44/43	0.008	14.3
31/23, 46/31	0.008	14.3
15/14, 28/15	0.009	15.0
21/13, 26/21	0.009	15.1
29/26, 52/29	0.009	15.1
13/10, 20/13	0.009	15.1
41/24, 48/41	0.009	15.2
17/16, 32/17	0.009	15.2
45/29, 58/45	0.009	15.2
9/8, 16/9	0.009	15.2
53/43, 86/53	0.009	15.3
23/13, 26/23	0.009	15.5
45/23, 46/45	0.009	15.6
43/37, 74/43	0.009	16.0
35/19, 38/35	0.009	16.1
31/21, 42/31	0.010	16.3
35/22, 44/35	0.010	16.6
31/19, 38/31	0.010	17.0
45/41, 82/45	0.010	17.1
17/15, 30/17	0.010	17.1
29/24, 48/29	0.010	17.1
5/3, 6/5	0.010	17.1
41/26, 52/41	0.010	17.2
51/26, 52/51	0.010	17.2
27/26, 52/27	0.010	17.2
37/32, 64/37	0.010	17.3
31/22, 44/31	0.010	17.5
23/12, 24/23	0.010	17.5
47/33, 66/47	0.010	17.5
53/35, 70/53	0.010	17.6
57/55, 110/57	0.010	17.6
57/47, 94/57	0.011	18.0
57/56, 112/57	0.011	18.0
19/13, 26/19	0.011	18.1
45/38, 76/45	0.011	18.3
41/31, 62/41	0.011	18.3
51/31, 62/51	0.011	18.4
31/27, 54/31	0.011	18.4
53/31, 62/53	0.011	18.4
33/28, 56/33	0.011	18.5
49/48, 96/49	0.011	18.6
13/11, 22/13	0.011	18.6
45/44, 88/45	0.011	18.7
25/16, 32/25	0.011	19.0
37/30, 60/37	0.011	19.2
41/35, 70/41	0.011	19.2
51/35, 70/51	0.011	19.2
35/27, 54/35	0.011	19.3
43/42, 84/43	0.011	19.5
53/52, 104/53	0.011	19.6
53/45, 90/53	0.012	19.7
57/34, 68/57	0.012	20.1
19/12, 24/19	0.012	20.1
43/25, 50/43	0.012	20.3
49/45, 90/49	0.012	20.5
33/17, 34/33	0.012	20.6
49/26, 52/49	0.012	20.6
11/6, 12/11	0.012	20.6
21/16, 32/21	0.012	20.8
47/30, 60/47	0.012	21.0
43/41, 82/43	0.013	21.5
51/43, 86/51	0.013	21.5
43/27, 54/43	0.013	21.6
53/48, 96/53	0.013	21.6
49/31, 62/49	0.013	21.8
57/37, 74/57	0.013	22.2
55/32, 64/55	0.013	22.5
29/28, 56/29	0.013	22.6
7/5, 10/7	0.013	22.6
37/33, 66/37	0.013	22.6
39/29, 58/39	0.013	22.8
39/20, 40/39	0.013	22.8
41/32, 64/41	0.013	22.8
51/32, 64/51	0.013	22.8
27/16, 32/27	0.013	22.9
47/32, 64/47	0.013	22.9
23/14, 28/23	0.013	23.0
39/23, 46/39	0.013	23.1
31/25, 50/31	0.014	23.5
55/43, 86/55	0.014	23.8
47/43, 86/47	0.014	24.2
25/13, 26/25	0.014	24.7
41/30, 60/41	0.014	24.7
29/17, 34/29	0.014	24.7
17/10, 20/17	0.014	24.7
29/18, 36/29	0.014	24.8
9/5, 10/9	0.014	24.8
49/43, 86/49	0.015	25.0
23/17, 34/23	0.015	25.0
23/18, 36/23	0.015	25.1
19/14, 28/19	0.015	25.6
39/38, 76/39	0.015	25.8
11/7, 14/11	0.015	26.1
49/32, 64/49	0.015	26.3
39/22, 44/39	0.015	26.3
47/35, 70/47	0.015	26.5
25/24, 48/25	0.016	26.7
37/29, 58/37	0.016	26.8
37/20, 40/37	0.016	26.8
55/31, 62/55	0.016	27.0
53/28, 56/53	0.016	27.1
37/23, 46/37	0.016	27.1
53/39, 78/53	0.016	27.3
47/31, 62/47	0.016	27.3
57/41, 82/57	0.016	27.7
19/17, 34/19	0.016	27.7
19/18, 36/19	0.016	27.8
49/30, 60/49	0.016	28.2
55/52, 104/55	0.016	28.2
41/33, 66/41	0.016	28.2
17/11, 22/17	0.016	28.2
11/9, 18/11	0.016	28.3
47/26, 52/47	0.017	28.5
47/45, 90/47	0.017	28.6
53/34, 68/53	0.017	29.2
53/36, 72/53	0.017	29.2
37/19, 38/37	0.017	29.8
55/48, 96/55	0.018	30.1
29/21, 42/29	0.018	30.3
21/20, 40/21	0.018	30.3
37/22, 44/37	0.018	30.3
47/24, 48/47	0.018	30.5
23/21, 42/23	0.018	30.6
57/49, 98/57	0.018	31.2
53/37, 74/53	0.018	31.3
49/33, 66/49	0.018	31.6
25/14, 28/25	0.019	32.2
39/25, 50/39	0.019	32.3
41/29, 58/41	0.019	32.3
41/40, 80/41	0.019	32.3
51/29, 58/51	0.019	32.3
51/40, 80/51	0.019	32.3
29/27, 54/29	0.019	32.4
27/20, 40/27	0.019	32.4
41/23, 46/41	0.019	32.7
51/46, 92/51	0.019	32.7
27/23, 46/27	0.019	32.7
21/19, 38/21	0.019	33.3
21/11, 22/21	0.020	33.8
25/17, 34/25	0.020	34.2
25/18, 36/25	0.020	34.3
53/42, 84/53	0.020	34.8
41/38, 76/41	0.021	35.3
51/38, 76/51	0.021	35.3
27/19, 38/27	0.021	35.4
55/28, 56/55	0.021	35.7
49/29, 58/49	0.021	35.8
49/40, 80/49	0.021	35.8
55/39, 78/55	0.021	35.8
41/22, 44/41	0.021	35.8
51/44, 88/51	0.021	35.8
27/22, 44/27	0.021	35.9
47/28, 56/47	0.021	36.0
49/46, 92/49	0.021	36.1
47/39, 78/47	0.021	36.1
37/25, 50/37	0.021	36.3
53/41, 82/53	0.021	36.8
53/51, 102/53	0.021	36.8
53/27, 54/53	0.022	36.9
55/34, 68/55	0.022	37.7
55/36, 72/55	0.022	37.8
47/34, 68/47	0.022	38.1
47/36, 72/47	0.022	38.1
49/38, 76/49	0.023	38.8
49/44, 88/49	0.023	39.3
25/21, 42/25	0.023	39.8
55/37, 74/55	0.023	39.8
47/37, 74/47	0.023	40.2
53/49, 98/53	0.023	40.3
41/25, 50/41	0.024	41.8
51/50, 100/51	0.024	41.8
27/25, 50/27	0.024	41.9
55/42, 84/55	0.025	43.3
47/42, 84/47	0.025	43.6
49/25, 50/49	0.026	45.3
55/41, 82/55	0.026	45.3
55/51, 102/55	0.026	45.3
55/54, 108/55	0.026	45.4
47/41, 82/47	0.027	45.7
51/47, 94/51	0.027	45.7
47/27, 54/47	0.027	45.8
55/49, 98/55	0.028	48.8
49/47, 94/49	0.029	49.2

Todo: fix template The error seems to be due to the consistency only calculating to 43, while 20567edo has a higher consistency limit, causing an error.