Minimal consistent EDOs: Difference between revisions
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An [[EDO|edo]] N is [[ | An [[EDO|edo]] N is [[consistent]] with respect to a set of rational numbers s if the [[Patent_val|patent val]] mapping of every element of s is the nearest N-edo approximation. It is ''uniquely consistent'' if every element of s is mapped to a unique value. If the set s is the q [[Odd_limit|odd limit]], we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least consistent, and least uniquely consistent, edo for every odd number up to 135. | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | Odd limit | |||
! | Smallest consistent | |||
! | Smallest uniquely consistent | |||
|- | |- | ||
| | 1 | | | 1 | ||
| Line 180: | Line 180: | ||
|- | |- | ||
| | 87 | | | 87 | ||
| | 2901533 | |||
| | 2901533 | |||
|- | |||
| | 89 | |||
| | 2901533 | | | 2901533 | ||
| | 2901533 | | | 2901533 | ||
| Line 317: | Line 321: | ||
=OEIS integer sequences links= | =OEIS integer sequences links= | ||
* {{OEIS|A116474|Equal divisions of the octave with progressively increasing consistency levels | * {{OEIS|A116474|Equal divisions of the octave with progressively increasing consistency levels}} | ||
* {{OEIS|A116475|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|A116475|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|A117577|Equal divisions of the octave with nondecreasing consistency levels.}} | * {{OEIS|A117577|Equal divisions of the octave with nondecreasing consistency levels.}} | ||
* {{OEIS|A117578|Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit | * {{OEIS|A117578|Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit}} | ||
Revision as of 13:19, 17 June 2019
An edo N is consistent with respect to a set of rational numbers s if the patent val mapping of every element of s is the nearest N-edo approximation. It is uniquely consistent if every element of s is mapped to a unique value. If the set s is the q odd limit, we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least consistent, and least uniquely consistent, edo for every odd number up to 135.
| Odd limit | Smallest consistent | Smallest uniquely consistent |
|---|---|---|
| 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 |
| 137 | 5407372813 | 5407372813 |
| 139 | 5407372813 | 5407372813 |
| 141 | 5407372813 | 5407372813 |
| 143 | 5407372813 | 5407372813 |
| 145 | 5407372813 | 5407372813 |
| 147 | 5407372813 | 5407372813 |
| 149 | 5407372813 | 5407372813 |
| 151 | 5407372813 | 5407372813 |
| 153 | 5407372813 | 5407372813 |
| 155 | 5407372813 | 5407372813 |
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)