Minimal consistent EDOs: Difference between revisions
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Cleanup, and explain odd limits beyond 135 in plain text |
Tristanbay (talk | contribs) Added a note mentioning that the table excludes 0edo from consistent EDOs (it has no consistency limit because it's consistent in all odd-limits) |
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! Odd<br>Limit | ! Odd<br>Limit | ||
! Smallest<br>Consistent Edo | ! Smallest<br>Consistent Edo* | ||
! Smallest<br>Distinctly Consistent Edo | ! Smallest<br>Distinctly Consistent Edo | ||
|- | |- | ||
| Line 279: | Line 279: | ||
| 70910024 | | 70910024 | ||
|} | |} | ||
<nowiki>*</nowiki>apart from 0edo | |||
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. | 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. | ||
Revision as of 23:14, 27 April 2024
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. 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 |
|---|---|---|
| 1 | 1 | 1 |
| 3 | 1 | 3 |
| 5 | 3 | 9 |
| 7 | 4 | 27 |
| 9 | 5 | 41 |
| 11 | 22 | 58 |
| 13 | 26 | 87 |
| 15 | 29 | 111 |
| 17 | 58 | 149 |
| 19 | 80 | 217 |
| 21 | 94 | 282 |
| 23 | 94 | 282 |
| 25 | 282 | 388 |
| 27 | 282 | 388 |
| 29 | 282 | 1323 |
| 31 | 311 | 1600 |
| 33 | 311 | 1600 |
| 35 | 311 | 1600 |
| 37 | 311 | 1600 |
| 39 | 311 | 2554 |
| 41 | 311 | 2554 |
| 43 | 17461 | 17461 |
| 45 | 17461 | 17461 |
| 47 | 20567 | 20567 |
| 49 | 20567 | 20567 |
| 51 | 20567 | 20567 |
| 53 | 20567 | 20567 |
| 55 | 20567 | 20567 |
| 57 | 20567 | 20567 |
| 59 | 253389 | 253389 |
| 61 | 625534 | 625534 |
| 63 | 625534 | 625534 |
| 65 | 625534 | 625534 |
| 67 | 625534 | 625534 |
| 69 | 759630 | 759630 |
| 71 | 759630 | 759630 |
| 73 | 759630 | 759630 |
| 75 | 2157429 | 2157429 |
| 77 | 2157429 | 2157429 |
| 79 | 2901533 | 2901533 |
| 81 | 2901533 | 2901533 |
| 83 | 2901533 | 2901533 |
| 85 | 2901533 | 2901533 |
| 87 | 2901533 | 2901533 |
| 89 | 2901533 | 2901533 |
| 91 | 2901533 | 2901533 |
| 93 | 2901533 | 2901533 |
| 95 | 2901533 | 2901533 |
| 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
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)