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Minimal consistent EDOs: Difference between revisions

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An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''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 interval error|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.
{{Idiosyncratic terms}}
An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''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''{{idiosyncratic}} if its [[relative interval error|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 limits of {{nowrap|2<sup>''n''</sup> &minus; 1}} are '''highlighted'''.


{| class="wikitable right-all"
<onlyinclude>{| class="wikitable center-all"
|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit
|-
|-
! Odd<br>limit
! Odd<br>limit !! Smallest<br>consistent edo* !! Smallest distinctly<br>consistent edo !! Smallest purely<br>consistent edo* !! Smallest edo<br>consistent to<br>[[Consistency #Generalization|distance 2]]* !! Smallest edo<br>distinctly consistent<br>to distance 2
! Smallest<br>consistent edo*
|- style="font-weight: bold; background-color: #dddddd;"
! Smallest distinctly<br>consistent edo
| 1 || 1 || 1 || 1 || 1 || 1
! Smallest ''purely<br>consistent''** edo
|- style="font-weight: bold; background-color: #dddddd;"
| 3 || 1 || 3 || 2 || 2 || 3
|-
|-
| 1
| 5 || 3 || 9 || 3 || 3 || 12
| 1
|- style="font-weight: bold; background-color: #dddddd;"
| 1
| 7 || 4 || 27 || 10 || 31 || 31
| 1
|-
|-
| 3
| 9 || 5 || 41 || 41 || 41 || 41
| 1
| 3
| 2
|-
|-
| 5
| 11 || 22 || 58 || 41 || 72 || 72
| 3
| 9
| 5
|-
|-
| 7
| 13 || 26 || 87 || 46 || 270 || 270
| 4
|- style="font-weight: bold; background-color: #dddddd;"
| 27
| 15 || 29 || 111 || 87 || 494 || 494
| 10
|-
|-
| 9
| 17 || 58 || 149 || 311 || 3395 || 3395
| 5
| 41
| 41
|-
|-
| 11
| 19 || 80 || 217 || 311 || 8539 || 8539
| 22
| 58
| 41
|-
|-
| 13
| 21 || 94 || 282 || 311 || 8539 || 8539
| 26
| 87
| 46
|-
|-
| 15
| 23 || 94 || 282 || 311 || 16808 || 16808
| 29
| 111
| 87
|-
|-
| 17
| 25 || 282 || 388 || 311 || 16808 || 16808
| 58
| 149
| 311
|-
|-
| 19
| 27 || 282 || 388 || 311 || 16808 || 16808
| 80
| 217
| 311
|-
|-
| 21
| 29 || 282 || 1323 || 311 || 16808 || 16808
| 94
|- style="font-weight: bold; background-color: #dddddd;"
| 282
| 31 || 311 || 1600 || 311 || 16808 || 16808
| 311
|-
|-
| 23
| 33 || 311 || 1600 || 311 || 16808 || 16808
| 94
| 282
| 311
|-
|-
| 25
| 35 || 311 || 1600 || 311 || 16808 || 16808
| 282
| 388
| 311
|-
|-
| 27
| 37 || 311 || 1600 || 311 || 324296 || 324296
| 282
| 388
| 311
|-
|-
| 29
| 39 || 311 || 2554 || 311 || 2398629 || 2398629
| 282
| 1323
| 311
|-
|-
| 31
| 41 || 311 || 2554 || 311 || 19164767 || 19164767
| 311
| 1600
| 311
|-
|-
| 33
| 43 || 17461 || 17461 || 20567 || 19735901 || 19735901
| 311
| 1600
| 311
|-
|-
| 35
| 45 || 17461 || 17461 || 20567 || 19735901 || 19735901
| 311
| 1600
| 311
|-
|-
| 37
| 47 || 20567 || 20567 || 20567 || 152797015 || 152797015
| 311
| 1600
| 311
|-
|-
| 39
| 49 || 20567 || 20567 || 459944 ||  ||  
| 311
| 2554
| 311
|-
|-
| 41
| 51 || 20567 || 20567 || 459944 ||  ||  
| 311
| 2554
| 311
|-
|-
| 43
| 53 || 20567 || 20567 || 1705229 ||  ||
| 17461
| 17461
| 20567
|-
|-
| 45
| 55 || 20567 || 20567 || 1705229 ||  ||
| 17461
| 17461
| 20567
|-
|-
| 47
| 57 || 20567 || 20567 || 1705229 ||  ||
| 20567
| 20567
| 20567
|-
|-
| 49
| 59 || 253389 || 253389 || 3159811 ||  ||  
| 20567
| 20567
|  
|-
|-
| 51
| 61 || 625534 || 625534 || 3159811 ||  ||
| 20567
|- style="font-weight: bold; background-color: #dddddd;"
| 20567
| 63 || 625534 || 625534 || 3159811 ||  ||  
|  
|-
|-
| 53
| 65 || 625534 || 625534 || 3159811 ||  ||  
| 20567
| 20567
|  
|-
|-
| 55
| 67 || 625534 || 625534 || 7317929 ||  ||  
| 20567
| 20567
|  
|-
|-
| 57
| 69 || 759630 || 759630 || 8595351 ||  ||  
| 20567
| 20567
|  
|-
|-
| 59
| 71 || 759630 || 759630 || 8595351 ||  ||  
| 253389
| 253389
|  
|-
|-
| 61
| 73 || 759630 || 759630 || 27783092 ||  ||  
| 625534
| 625534
|  
|-
|-
| 63
| 75 || 2157429 || 2157429 || 34531581 ||  ||  
| 625534
| 625534
| 3159811
|-
|-
| 65
| 77 || 2157429 || 2157429 || 34531581 ||  ||  
| 625534
| 625534
|  
|-
|-
| 67
| 79 || 2901533 || 2901533 || 50203972 ||  ||  
| 625534
| 625534
|  
|-
|-
| 69
| 81 || 2901533 || 2901533 || 50203972 ||  ||  
| 759630
| 759630
|  
|-
|-
| 71
| 83 || 2901533 || 2901533 || 50203972 ||  ||  
| 759630
| 759630
|  
|-
|-
| 73
| 85 || 2901533 || 2901533 || 50203972 ||  ||  
| 759630
| 759630
|  
|-
|-
| 75
| 87 || 2901533 || 2901533 || 50203972 ||  ||  
| 2157429
| 2157429
|  
|-
|-
| 77
| 89 || 2901533 || 2901533 || 50203972 ||  ||  
| 2157429
| 2157429
|  
|-
|-
| 79
| 91 || 2901533 || 2901533 || 50203972 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 81
| 93 || 2901533 || 2901533 || 50203972 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 83
| 95 || 2901533 || 2901533 || 50203972 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 85
| 97 || 2901533 || 2901533 || 1297643131 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 87
| 99 || 2901533 || 2901533 || 1297643131 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 89
| 101 || 2901533 || 2901533 || 3888109922 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 91
| 103 || 2901533 || 2901533 || 3888109922 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 93
| 105 || 2901533 || 2901533 || 3888109922 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 95
| 107 || 2901533 || 2901533 || 13805152233 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 97
| 109 || 2901533 || 2901533 || 27218556026 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 99
| 111 || 2901533 || 2901533 || 27218556026 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 101
| 113 || 2901533 || 2901533 || 27218556026 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 103
| 115 || 2901533 || 2901533 || 27218556026 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 105
| 117 || 2901533 || 2901533 || 27218556026 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 107
| 119 || 2901533 || 2901533 || 42586208631 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 109
| 121 || 2901533 || 2901533 || 42586208631 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 111
| 123 || 2901533 || 2901533 || 42586208631 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 113
| 125 || 2901533 || 2901533 || 42586208631 ||  ||
| 2901533
|- style="font-weight: bold; background-color: #dddddd;"
| 2901533
| 127 || 2901533 || 2901533 || 42586208631 ||  ||  
|  
|-
|-
| 115
| 129 || 2901533 || 2901533 || 42586208631 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 117
| 131 || 2901533 || 2901533 || 93678217813** ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 119
| 133 || 70910024 || 70910024 || 93678217813 ||  ||  
| 2901533
| 2901533
|  
|-
|-
| 121
| 135 || 70910024 || 70910024 || 93678217813 || ||
| 2901533
| 2901533
|
|-
| 123
| 2901533
| 2901533
|
|-
| 125
| 2901533
| 2901533
|
|-
| 127
| 2901533
| 2901533
|
|-
| 129
| 2901533
| 2901533
|
|-
| 131
| 2901533
| 2901533
|
|-
| 133
| 70910024
| 70910024
|  
|-
| 135
| 70910024
| 70910024
|  
|}
|}
<nowiki>*</nowiki> apart from 0edo
<nowiki />* Apart from 0edo


<nowiki>**</nowiki> ''purely consistent'' is an {{idiosyncratic}}
<nowiki />** Purely consistent to the 137-odd-limit</onlyinclude>


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 [[5407372813edo|5407372813]], reported to be consistent to the 155-odd-limit.


== 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}}
== See also ==
* [[Consistency limits of small EDOs]]
* {{u|ArrowHead294|Purely consistent EDOs by odd limit}}


[[Category:Mapping]]
[[Category:Mapping]]
[[Category:Consistency]]
[[Category:Consistency]]
[[Category:Odd limit]]
[[Category:Odd limit]]
Retrieved from "https://en.xen.wiki/w/Minimal_consistent_EDOs"