Minimal consistent EDOs
An edo N is consistent with respect to the q-odd-limit if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics. It is distinctly consistent if every one of those closest approximations is a distinct value, and purely consistent if its relative errors on odd harmonics up to and including q never exceed 25%. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135.
Odd limit |
Smallest consistent edo* |
Smallest distinctly consistent edo |
Smallest purely consistent** edo |
---|---|---|---|
1 | 1 | 1 | 1 |
3 | 1 | 3 | 2 |
5 | 3 | 9 | 3 |
7 | 4 | 27 | 10 |
9 | 5 | 41 | 41 |
11 | 22 | 58 | 41 |
13 | 26 | 87 | 46 |
15 | 29 | 111 | 87 |
17 | 58 | 149 | 311 |
19 | 80 | 217 | 311 |
21 | 94 | 282 | 311 |
23 | 94 | 282 | 311 |
25 | 282 | 388 | 311 |
27 | 282 | 388 | 311 |
29 | 282 | 1323 | 311 |
31 | 311 | 1600 | 311 |
33 | 311 | 1600 | 311 |
35 | 311 | 1600 | 311 |
37 | 311 | 1600 | 311 |
39 | 311 | 2554 | 311 |
41 | 311 | 2554 | 311 |
43 | 17461 | 17461 | 20567 |
45 | 17461 | 17461 | 20567 |
47 | 20567 | 20567 | 20567 |
49 | 20567 | 20567 | 459944 |
51 | 20567 | 20567 | 459944 |
53 | 20567 | 20567 | 1705229 |
55 | 20567 | 20567 | 1705229 |
57 | 20567 | 20567 | 1705229 |
59 | 253389 | 253389 | 3159811 |
61 | 625534 | 625534 | 3159811 |
63 | 625534 | 625534 | 3159811 |
65 | 625534 | 625534 | 3159811 |
67 | 625534 | 625534 | 7317929 |
69 | 759630 | 759630 | 8595351 |
71 | 759630 | 759630 | 8595351 |
73 | 759630 | 759630 | 27783092 |
75 | 2157429 | 2157429 | 34531581 |
77 | 2157429 | 2157429 | 34531581 |
79 | 2901533 | 2901533 | 50203972 |
81 | 2901533 | 2901533 | 50203972 |
83 | 2901533 | 2901533 | 50203972 |
85 | 2901533 | 2901533 | 50203972 |
87 | 2901533 | 2901533 | 50203972 |
89 | 2901533 | 2901533 | 50203972 |
91 | 2901533 | 2901533 | 50203972 |
93 | 2901533 | 2901533 | 50203972 |
95 | 2901533 | 2901533 | 50203972 |
97 | 2901533 | 2901533 | |
99 | 2901533 | 2901533 | |
101 | 2901533 | 2901533 | |
103 | 2901533 | 2901533 | |
105 | 2901533 | 2901533 | |
107 | 2901533 | 2901533 | |
109 | 2901533 | 2901533 | |
111 | 2901533 | 2901533 | |
113 | 2901533 | 2901533 | |
115 | 2901533 | 2901533 | |
117 | 2901533 | 2901533 | |
119 | 2901533 | 2901533 | |
121 | 2901533 | 2901533 | |
123 | 2901533 | 2901533 | |
125 | 2901533 | 2901533 | |
127 | 2901533 | 2901533 | |
129 | 2901533 | 2901533 | |
131 | 2901533 | 2901533 | |
133 | 70910024 | 70910024 | |
135 | 70910024 | 70910024 |
* apart from 0edo
** purely consistent is an [idiosyncratic term]
The last entry, 70910024edo, is consistent up to the 135-odd-limit. The next edo is 5407372813, reported to be consistent to the 155-odd-limit.
OEIS integer sequences links
- OEIS: Equal divisions of the octave with progressively increasing consistency levels (OEIS)
- OEIS: Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit (OEIS)
- OEIS: Equal divisions of the octave with nondecreasing consistency levels. (OEIS)
- OEIS: Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit (OEIS)