Relative Errors of Small Edos

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The relative error of an interval in an edo is the error approximating JI divided by the size of a single step. The formula for closest mapping: error(n, r) = round (n log2r) - n log2r, where n is the edo number and r is the frequency ratio. With closest mapping, the relative error ranges from -0.5 to +0.5. With patent val mapping, it can be farther from zero.

This article contains two lists. The first shows relative errors of the first 9 prime harmonies 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 (see below), 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 the criterion.

Additivity

There are two additivities of relative errors.

First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if r3 = r1r2 for n, then error (n, r3) = error (n, r1) + error (n, r2).

If the error exceeds the range -0.5 to +0.5, it indicates that an inconsistency occurs, and there is a discrepancy in patent val mapping and closest mapping, so is the error. The patent val mapping error is unchanged, and that of closest mapping is the previous result reduced by an integer to fit it into the range.

For example, the errors of 2/1, 3/1 and 5/1 in 19-edo are 0, -0.1143 and -0.1166, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-0.1143) - (-0.1166) = 0.0023. That shows 19-edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

Second, for the same ratio, an edo which is the sum of some other edos have their relative errors additive, that is, if n3 = n1 + n2 for r, then error (n3, r) = error (n1, r) + error (n2, r). This too needs to be reduced by an integer to fit into the range. Specially, if an edo duplicates itself, and if the mappings do not change, then the error also duplicates.

For example, the errors of 3/1 for 26-edo and 27-edo are -0.2090 and +0.2060, repectively, and their sum -0.0030 is the error of 3/1 for 53-edo.

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