Relative errors of small EDOs

This article contains two lists. The first shows relative errors of the first 9 prime harmonics for edos up to 99. There is no point for showing higher primes because no edo under 99 is consistent up to them. For other intervals, the relative error follows the additive rule, so they can be derived easily. Also by that rule, however, finding large errors with such p/1 harmonies will not suffice that the edo does a poor approximation in the p-limit overall. One must inspect every relevant interval to be sure of that. The second list comes in naturally for showing the root-mean-squared relative errors of a certain JI subgroup, and may be used as a criterion.

List of Relative Errors of Prime Harmonies for Small Edos

Edo	Relative Errors (Percent)
	2/1	3/1	5/1	7/1	11/1	13/1	17/1	19/1	23/1
1	0.00	41.50	-32.19	19.26	-45.94	29.96	-8.75	-24.79	47.64
2	0.00	-16.99	35.61	38.53	8.11	-40.09	-17.49	-49.59	-4.71
3	0.00	24.51	3.42	-42.21	-37.83	-10.13	-26.24	25.62	42.93
4	0.00	-33.99	-28.77	-22.94	16.23	19.82	-34.99	0.83	-9.42
5	0.00	7.52	39.04	-3.68	-29.72	49.78	-43.73	-23.96	38.22
6	0.00	49.02	6.84	15.59	24.34	-20.26	47.52	-48.76	-14.14
7	0.00	-9.47	-25.35	34.85	-21.60	9.69	38.78	26.45	33.51
8	0.00	32.03	42.46	-45.88	32.45	39.65	30.03	1.66	-18.85
9	0.00	-26.47	10.26	-26.62	-13.49	-30.40	21.28	-23.13	28.79
10	0.00	15.04	-21.93	-7.35	40.57	-0.44	12.54	-47.93	-23.56
11	0.00	-43.46	45.88	11.91	-5.37	29.52	3.79	27.28	24.08
12	0.00	-1.96	13.69	31.17	48.68	-40.53	-4.96	2.49	-28.27
13	0.00	39.55	-18.51	-49.56	2.74	-10.57	-13.70	-22.31	19.37
14	0.00	-18.95	49.30	-30.30	-43.20	19.38	-22.45	-47.10	-32.99
15	0.00	22.56	17.11	-11.03	10.85	49.34	-31.19	28.11	14.66
16	0.00	-35.94	-15.08	8.23	-35.09	-20.70	-39.94	3.32	-37.70
17	0.00	5.56	-47.28	27.50	18.97	9.25	-48.69	-21.48	9.94
18	0.00	47.07	20.53	46.76	-26.98	39.21	42.57	-46.27	-42.41
19	0.00	-11.43	-11.66	-33.97	27.08	-30.84	33.82	28.94	5.23
20	0.00	30.07	-43.86	-14.71	-18.86	-0.88	25.07	4.14	-47.12
21	0.00	-28.42	23.95	4.55	35.19	29.08	16.33	-20.65	0.52
22	0.00	13.08	-8.24	23.82	-10.75	-40.97	7.58	-45.44	48.16
23	0.00	-45.41	-40.43	43.08	43.31	-11.01	-1.16	29.77	-4.19
24	0.00	-3.91	27.37	-37.65	-2.64	18.94	-9.91	4.97	43.45
25	0.00	37.59	-4.82	-18.39	-48.58	48.90	-18.66	-19.82	-8.90
26	0.00	-20.90	-37.01	0.88	5.48	-21.14	-27.40	-44.61	38.74
27	0.00	20.60	30.79	20.14	-40.47	8.81	-36.15	30.60	-13.62
28	0.00	-37.90	-1.40	39.41	13.59	38.77	-44.90	5.80	34.03
29	0.00	3.61	-33.59	-41.33	-32.35	-31.28	46.36	-18.99	-18.33
30	0.00	45.11	34.22	-22.06	21.71	-1.32	37.61	-43.78	29.31
31	0.00	-13.38	2.02	-2.80	-24.24	28.64	28.87	31.42	-23.04
32	0.00	28.12	-30.17	16.46	29.82	-41.41	20.12	6.63	24.60
33	0.00	-30.38	37.64	35.73	-16.12	-11.45	11.37	-18.16	-27.75
34	0.00	11.13	5.44	-45.01	37.93	18.50	2.63	-42.95	19.89
35	0.00	-47.37	-26.75	-25.74	-8.01	48.46	-6.12	32.25	-32.47
36	0.00	-5.87	41.06	-6.48	46.05	-21.58	-14.87	7.46	15.18
37	0.00	35.64	8.87	12.79	0.10	8.37	-23.61	-17.33	-37.18
38	0.00	-22.86	-23.33	32.05	-45.84	38.33	-32.36	-42.12	10.46
39	0.00	18.65	44.48	-48.68	8.22	-31.71	-41.11	33.08	-41.89
40	0.00	-39.85	12.29	-29.42	-37.73	-1.76	-49.85	8.29	5.75
41	0.00	1.65	-19.91	-10.16	16.33	28.20	41.40	-16.50	-46.60
42	0.00	43.16	47.90	9.11	-29.61	-41.85	32.66	-41.30	1.04
43	0.00	-15.34	15.71	28.37	24.44	-11.89	23.91	33.91	48.68
44	0.00	26.16	-16.48	47.64	-21.50	18.07	15.16	9.12	-3.67
45	0.00	-32.33	-48.68	-33.10	32.56	48.02	6.42	-15.67	43.97
46	0.00	9.17	19.13	-13.83	-13.39	-22.02	-2.33	-40.47	-8.38
47	0.00	-49.32	-13.06	5.43	40.67	7.93	-11.08	34.74	39.26
48	0.00	-7.82	-45.25	24.70	-5.27	37.89	-19.82	9.95	-13.10
49	0.00	33.68	22.55	43.96	48.79	-32.15	-28.57	-14.84	34.55
50	0.00	-24.81	-9.64	-36.77	2.84	-2.20	-37.31	-39.64	-17.81
51	0.00	16.69	-41.83	-17.51	-43.10	27.76	-46.06	35.57	29.83
52	0.00	-41.81	25.97	1.75	10.96	-42.29	45.19	10.78	-22.52
53	0.00	-0.30	-6.22	21.02	-34.99	-12.33	36.45	-14.02	25.12
54	0.00	41.20	-38.41	40.28	19.07	17.63	27.70	-38.81	-27.23
55	0.00	-17.29	29.40	-40.45	-26.87	47.58	18.95	36.40	20.41
56	0.00	24.21	-2.80	-21.19	27.18	-22.46	10.21	11.61	-31.95
57	0.00	-34.29	-34.99	-1.92	-18.76	7.49	1.46	-13.19	15.70
58	0.00	7.22	32.82	17.34	35.30	37.45	-7.28	-37.98	-36.66
59	0.00	48.72	0.62	36.61	-10.65	-32.59	-16.03	37.23	10.98
60	0.00	-9.78	-31.57	-44.13	43.41	-2.64	-24.78	12.43	-41.37
61	0.00	31.73	36.24	-24.87	-2.53	27.32	-33.52	-12.36	6.27
62	0.00	-26.77	4.05	-5.60	-48.48	-42.73	-42.27	-37.15	-46.08
63	0.00	14.74	-28.15	13.66	5.58	-12.77	48.98	38.06	1.56
64	0.00	-43.76	39.66	32.93	-40.36	17.19	40.24	13.26	49.20
65	0.00	-2.26	7.47	-47.81	13.69	47.14	31.49	-11.53	-3.15
66	0.00	39.25	-24.73	-28.54	-32.25	-22.90	22.75	-36.32	44.49
67	0.00	-19.25	43.08	-9.28	21.81	7.05	14.00	38.89	-7.87
68	0.00	22.25	10.89	9.99	-24.14	37.01	5.25	14.09	39.78
69	0.00	-36.24	-21.30	29.25	29.92	-33.03	-3.49	-10.70	-12.58
70	0.00	5.26	46.50	48.52	-16.02	-3.08	-12.24	-35.49	35.07
71	0.00	46.77	14.31	-32.22	38.04	26.88	-20.99	39.71	-17.29
72	0.00	-11.73	-17.88	-12.96	-7.91	-43.17	-29.73	14.92	30.35
73	0.00	29.77	49.92	6.31	46.15	-13.21	-38.48	-9.87	-22.00
74	0.00	-28.72	17.73	25.57	0.21	16.75	-47.23	-34.66	25.64
75	0.00	12.78	-14.46	44.84	-45.74	46.70	44.03	40.54	-26.71
76	0.00	-45.72	-46.65	-35.90	8.32	-23.34	35.28	15.75	20.93
77	0.00	-4.21	21.15	-16.63	-37.62	6.61	26.54	-9.04	-31.43
78	0.00	37.29	-11.04	2.63	16.43	36.57	17.79	-33.83	16.22
79	0.00	-21.20	-43.23	21.90	-29.51	-33.47	9.04	41.37	-36.14
80	0.00	20.30	24.58	41.16	24.55	-3.52	0.30	16.58	11.50
81	0.00	-38.20	-7.62	-39.57	-21.40	26.44	-8.45	-8.21	-40.85
82	0.00	3.31	-39.81	-20.31	32.66	-43.61	-17.20	-33.01	6.79
83	0.00	44.81	28.00	-1.05	-13.28	-13.65	-25.94	42.20	-45.56
84	0.00	-13.69	-4.20	18.22	40.77	16.31	-34.69	17.41	2.08
85	0.00	27.82	-36.39	37.48	-5.17	46.26	-43.43	-7.38	49.72
86	0.00	-30.68	31.42	-43.25	48.89	-23.78	47.82	-32.18	-2.63
87	0.00	10.83	-0.77	-23.99	2.94	6.17	39.07	43.03	45.01
88	0.00	-47.67	-32.97	-4.72	-43.00	36.13	30.33	18.24	-7.35
89	0.00	-6.17	34.84	14.54	11.06	-33.91	21.58	-6.55	40.30
90	0.00	35.34	2.65	33.81	-34.88	-3.96	12.83	-31.35	-12.06
91	0.00	-23.16	-29.55	-46.93	19.17	26.00	4.09	43.86	35.59
92	0.00	18.34	38.26	-27.67	-26.77	-44.05	-4.66	19.07	-16.77
93	0.00	-40.15	6.07	-8.40	27.29	-14.09	-13.40	-5.73	30.87
94	0.00	1.35	-26.12	10.86	-18.66	15.87	-22.15	-30.52	-21.48
95	0.00	42.86	41.68	30.13	35.40	45.82	-30.90	44.69	26.16
96	0.00	-15.64	9.49	49.39	-10.54	-24.22	-39.64	19.90	-26.19
97	0.00	25.86	-22.70	-31.34	43.51	5.73	-48.39	-4.90	21.45
98	0.00	-32.63	45.10	-12.08	-2.43	35.69	42.86	-29.69	-30.91
99	0.00	8.87	12.91	7.19	-48.37	-34.35	34.12	45.52	16.74

List of RMS Relative Errors of JI Subgroups for Small Edos

Notes:

1. Patent val mapping is used throughout.
2. Octave equivalence is assumed. For example, the 5-odd-limit takes account of 1/1, 3/1, 5/1, 5/3 and their octave inverses.
3. The weight of intervals is determined by the tonality diamond. For example, the 9-odd-limit takes account of 3/1 twice.

Edo	Root Mean Squared Relative Errors (Percent)
	3 Odd Limit	5 Odd Limit	7 Odd Limit	9 Odd Limit	11 Odd Limit	13 Odd Limit	15 Odd Limit	17 Odd Limit	19 Odd Limit	21 Odd Limit	23 Odd Limit
1	29.35	42.66	38.25	54.88	61.68	57.87	54.17	51.99	51.57	53.79	53.35
2	12.02	31.00	33.35	40.44	36.97	40.91	39.38	38.05	41.60	41.18	39.43
3	17.33	15.33	34.23	42.69	45.56	42.45	42.08	41.59	41.08	39.93	41.80
4	24.03	21.13	18.34	31.02	37.61	40.74	43.75	41.62	40.86	41.54	40.21
5	5.32	23.91	23.81	21.44	29.27	35.10	36.78	43.55	43.62	41.59	41.82
6	34.66	30.66	26.61	51.22	47.03	50.72	49.62	47.53	56.32	56.32	56.23
7	6.70	14.79	31.20	29.90	28.86	27.93	29.75	34.97	35.45	35.34	35.97
8	22.65	25.55	48.63	54.11	49.94	47.16	50.14	47.27	46.42	47.29	48.19
9	18.71	21.88	22.98	31.51	28.92	27.45	25.73	30.28	28.94	31.48	35.50
10	10.63	21.47	18.85	25.42	30.45	28.61	27.65	26.22	34.04	32.56	32.73
11	30.73	51.59	45.19	65.23	59.74	59.17	55.52	52.52	51.64	50.73	49.81
12	1.38	9.84	18.74	18.70	27.49	37.31	34.98	33.43	31.75	31.76	33.51
13	27.97	34.24	45.60	63.59	58.18	54.71	51.65	49.64	48.56	46.59	45.14
14	13.40	40.68	43.05	44.08	44.41	44.22	45.23	43.44	44.75	45.54	44.25
15	15.95	13.59	18.95	27.35	25.06	29.02	28.80	35.91	34.47	32.94	31.54
16	25.41	20.84	23.76	40.61	37.62	34.90	34.95	33.39	34.53	32.93	31.86
17	3.93	33.53	38.55	35.46	33.98	31.63	36.39	40.08	38.50	40.44	39.15
18	33.28	27.25	27.87	44.68	54.53	50.68	50.32	47.53	57.22	60.69	65.40
19	8.08	7.70	17.42	16.32	27.15	27.40	26.15	32.36	34.49	36.87	35.68
20	21.27	42.93	37.69	50.84	48.27	44.72	42.45	41.57	39.45	37.96	41.45
21	20.10	30.27	26.48	39.94	43.95	43.78	41.04	39.33	38.55	38.00	36.40
22	9.25	12.42	17.34	18.87	20.68	30.63	28.72	27.24	32.68	35.13	38.42
23	32.11	28.76	50.48	63.98	69.09	64.00	68.54	65.37	65.64	62.83	60.29
24	2.76	19.68	32.65	29.30	26.76	27.28	27.93	26.88	25.56	29.95	33.86
25	26.58	26.81	29.35	48.26	56.30	56.12	53.32	52.49	51.55	49.34	47.78
26	14.78	21.43	22.30	25.31	26.66	24.84	30.29	28.68	28.82	27.54	36.02
27	14.57	18.11	15.80	19.33	37.61	34.85	37.55	42.45	41.14	40.85	40.71
28	26.80	24.81	38.67	55.10	52.53	54.37	53.56	53.12	50.95	48.81	49.68
29	2.55	23.69	28.09	28.84	28.27	27.23	26.06	38.29	36.43	36.16	34.64
30	31.90	27.18	37.90	54.71	50.11	48.64	52.35	49.44	56.78	54.14	51.88
31	9.46	9.67	8.39	15.13	16.20	24.64	23.22	26.82	29.15	28.68	28.86
32	19.88	33.66	30.98	40.68	38.04	45.48	42.83	40.79	38.70	39.73	38.37
33	21.48	39.34	39.71	53.83	49.58	45.99	43.50	41.64	40.02	38.37	37.93
34	7.87	6.42	31.44	32.66	36.26	34.20	32.30	30.55	36.09	37.48	36.77
35	33.49	27.42	23.76	44.83	44.04	57.61	59.14	56.64	59.15	60.36	57.86
36	4.15	29.52	27.89	27.14	33.46	34.81	35.32	35.03	33.23	32.64	31.49
37	25.20	21.42	18.60	36.26	35.75	33.72	33.63	37.80	38.79	39.54	43.41
38	16.16	15.40	31.97	37.24	38.40	45.06	45.49	43.64	42.83	42.40	41.91
39	13.18	25.79	48.19	47.12	43.03	44.87	50.25	52.82	51.45	51.48	52.56
40	28.18	31.47	29.94	45.70	42.07	41.23	38.58	37.96	39.19	41.56	41.99
41	1.17	13.85	12.25	12.44	15.99	20.85	21.84	27.57	27.66	26.76	31.80
42	30.52	30.48	29.43	43.52	53.12	59.59	65.77	62.08	65.55	63.88	61.75
43	10.85	17.93	23.24	29.85	30.23	29.04	27.17	27.50	29.04	27.80	31.12
44	18.50	24.83	34.67	37.73	41.36	38.33	35.94	33.89	32.23	39.18	38.59
45	22.86	28.60	25.05	30.33	45.42	55.24	60.43	58.39	55.42	56.13	59.79
46	6.49	11.05	17.16	17.47	19.10	21.65	24.29	23.04	28.49	27.19	26.15
47	34.88	29.51	30.18	54.85	62.76	59.63	59.97	56.72	58.62	57.15	59.00
48	5.53	27.93	35.48	32.10	29.36	35.49	41.00	39.01	37.90	37.47	36.00
49	23.82	19.82	23.09	31.61	29.96	43.69	43.20	48.35	48.67	52.04	49.97
50	17.55	14.44	19.91	25.32	27.15	26.58	26.14	25.91	25.78	29.12	28.10
51	11.80	34.81	30.66	37.05	40.21	41.37	40.00	41.66	44.12	42.12	42.63
52	29.56	39.49	34.47	54.80	52.54	50.56	47.31	52.73	50.88	50.14	48.19
53	0.21	4.05	14.63	13.30	23.32	22.02	20.62	26.78	26.05	26.47	27.14
54	29.13	45.97	46.54	58.09	53.13	49.29	47.08	44.50	49.35	54.18	55.25
55	12.23	27.26	36.05	35.80	33.53	43.84	41.88	40.75	41.83	46.81	45.74
56	17.12	17.15	22.83	34.14	32.50	34.74	33.13	31.24	29.65	28.40	31.56
57	24.24	23.10	23.84	35.77	33.01	34.87	39.51	39.42	37.70	36.33	38.15
58	5.10	19.91	17.42	15.58	18.00	19.28	20.19	23.33	33.00	31.76	36.48
59	34.45	32.27	30.56	50.92	52.75	57.52	55.52	55.24	52.96	56.85	54.79
60	6.91	18.66	24.63	22.05	39.47	36.74	37.51	35.74	35.78	38.17	37.90
61	22.44	22.80	35.17	43.33	41.50	38.74	42.93	48.09	47.45	45.38	43.57
62	18.93	19.34	16.79	30.27	32.40	31.75	29.71	29.11	27.98	26.75	26.58
63	10.42	25.16	24.46	27.48	25.09	24.99	24.64	31.49	32.75	32.14	31.01
64	30.94	48.18	46.71	67.95	63.84	61.41	57.55	58.80	56.33	53.79	55.53
65	1.60	5.88	30.76	27.73	28.10	36.84	34.49	35.00	33.99	39.13	37.48
66	27.75	37.25	38.19	57.61	57.72	55.24	52.04	49.84	50.46	48.27	49.12
67	13.61	36.85	33.61	38.37	37.73	35.12	34.59	32.93	34.44	36.06	34.88
68	15.74	12.85	11.15	21.53	29.47	30.25	29.63	28.40	26.95	26.60	27.00
69	25.63	21.03	34.80	48.35	51.43	48.76	49.79	47.53	45.20	43.31	41.50
70	3.72	29.41	31.86	29.66	33.73	32.60	35.45	36.02	40.23	42.36	41.42
71	33.07	27.67	40.14	60.61	55.78	51.65	50.91	53.28	50.91	48.80	49.64
72	8.29	10.49	9.28	11.02	10.44	18.07	17.94	17.66	22.25	21.50	27.66
73	21.05	29.00	28.01	33.03	31.46	36.71	42.68	51.07	50.59	48.62	49.68
74	20.31	27.07	29.50	43.48	39.96	38.83	36.48	39.27	38.77	37.14	38.34
75	9.04	15.74	30.96	28.93	40.99	43.48	41.03	41.78	41.39	43.06	44.95
76	32.33	30.79	26.85	41.29	46.68	43.62	49.21	57.43	58.19	59.67	60.83
77	2.98	15.70	19.26	17.88	25.03	24.22	24.99	26.92	25.86	26.01	27.38
78	26.37	29.24	25.60	44.46	40.65	38.52	36.07	34.09	40.15	39.44	37.77
79	14.99	24.96	34.25	35.51	33.09	31.45	35.72	37.20	44.52	43.00	42.02
80	14.35	15.16	20.73	21.46	19.59	23.04	24.26	25.19	23.99	28.19	27.47
81	27.01	23.34	25.10	38.39	35.52	43.31	41.97	40.27	38.75	43.37	41.87
82	2.34	27.71	24.51	24.88	31.98	35.80	35.92	33.94	33.32	31.78	31.60
83	31.69	26.14	27.44	47.40	49.48	49.59	52.72	54.79	52.81	51.10	56.03
84	9.68	8.10	16.38	21.35	31.18	29.69	29.65	32.25	31.74	30.33	29.05
85	19.67	37.18	40.58	45.47	43.11	43.13	42.21	47.43	45.48	49.49	49.63
86	21.69	35.85	40.91	46.50	56.12	52.46	49.39	53.13	51.87	56.60	54.32
87	7.66	7.49	17.95	21.45	19.59	18.27	17.46	22.87	26.46	27.06	29.54
88	33.71	28.19	27.93	48.75	44.65	55.03	57.31	61.03	60.92	59.39	57.26
89	4.36	25.53	22.32	23.87	21.94	28.68	29.76	29.05	28.11	26.83	29.08
90	24.99	22.73	23.56	36.52	47.19	45.01	43.48	41.05	44.05	47.93	47.32
91	16.38	17.95	23.77	24.47	33.89	39.10	40.72	39.76	45.62	49.78	51.74
92	12.97	22.10	34.33	34.93	38.20	43.31	48.59	46.07	44.10	42.54	41.62
93	28.39	29.00	25.18	45.40	49.64	45.97	43.85	41.39	39.58	39.96	42.98
94	0.96	17.88	19.42	17.71	18.34	19.87	21.42	21.61	22.84	23.45	23.13
95	30.30	28.19	24.44	38.98	35.67	33.18	37.32	48.98	46.57	46.63	45.03
96	11.06	14.65	33.96	38.60	35.89	35.36	33.10	35.08	35.27	37.11	36.83
97	18.29	28.06	31.37	43.50	44.67	41.44	38.92	44.81	42.64	40.78	39.81
98	23.07	45.07	40.28	51.63	47.45	49.69	47.15	48.80	48.28	49.40	48.41
99	6.27	7.63	6.60	8.38	31.37	33.58	33.84	35.67	38.51	36.92	35.54

See Also

• EDO
• Relative error
• Relative Errors of Small EDTs
• Relative Errors of Small EDFs
• Relative Errors of Small ED6s