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Minimal consistent EDOs: Difference between revisions

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An [[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 smallest consistent, and the smallest uniquely consistent, edo for every odd number up to 135.
{{Idiosyncratic terms|The term ‘purely consistent’ was coined by [[User:ArrowHead294]] and does not so far see widespread use.}}
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. Odd limits of {{nowrap|2<sup>''n''</sup> &minus; 1}} are '''highlighted'''.


{| class="wikitable"
<onlyinclude>
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit
|-
|-
! Odd 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 consistent edo
|- style="font-weight: bold; background-color: #dddddd;"
! Smallest uniquely consistent edo
| 1 || [[1edo|1]]|| 1 || 1 || 1 || 1
|- style="font-weight: bold; background-color: #dddddd;"
| 3 || 1 || [[3edo|3]]|| [[2edo|2]]|| 2 || 3
|-
|-
| 1
| 5 || [[3edo|3]]|| [[9edo|9]]|| 3 || 3 || [[12edo|12]]
| 1
|- style="font-weight: bold; background-color: #dddddd;"
| 1
| 7 || [[4edo|4]]|| [[27edo|27]]|| [[10edo|10]]|| [[31edo|31]]|| 31
|-
|-
| 3
| 9 || [[5edo|5]]|| [[41edo|41]]|| 41 || 41 || 41
| 1
| 3
|-
|-
| 5
| 11 || [[22edo|22]]|| [[58edo|58]]|| 41 || [[72edo|72]]|| 72
| 3
| 9
|-
|-
| 7
| 13 || [[26edo|26]]|| [[87edo|87]]|| [[46edo|46]]|| [[270edo|270]]|| 270
| 4
|- style="font-weight: bold; background-color: #dddddd;"
| 27
| 15 || [[29edo|29]]|| [[111edo|111]]|| [[87edo|87]]|| [[494edo|494]]|| 494
|-
|-
| 9
| 17 || [[58edo|58]]|| [[149edo|149]]|| [[311edo|311]]|| [[3395edo|3395]]|| 3395
| 5
| 41
|-
|-
| 11
| 19 || [[80edo|80]]|| [[217edo|217]]|| 311 || [[8539edo|8539]]|| 8539
| 22
| 58
|-
|-
| 13
| 21 || [[94edo|94]]|| [[282edo|282]]|| 311 || 8539 || 8539
| 26
| 87
|-
|-
| 15
| 23 || 94 || 282 || 311 || [[16808edo|16808]]|| 16808
| 29
| 111
|-
|-
| 17
| 25 || [[282edo|282]]|| [[388edo|388]]|| 311 || 16808 || 16808
| 58
| 149
|-
|-
| 19
| 27 || 282 || 388 || 311 || 16808 || 16808
| 80
| 217
|-
|-
| 21
| 29 || 282 || [[1323edo|1323]]|| 311 || 16808 || 16808
| 94
|- style="font-weight: bold; background-color: #dddddd;"
| 282
| 31 || [[311edo|311]]|| [[1600edo|1600]]|| 311 || 16808 || 16808
|-
|-
| 23
| 33 || 311 || 1600 || 311 || 16808 || 16808
| 94
| 282
|-
|-
| 25
| 35 || 311 || 1600 || 311 || 16808 || 16808
| 282
| 388
|-
|-
| 27
| 37 || 311 || 1600 || 311 || [[324296edo|324296]]|| 324296
| 282
| 388
|-
|-
| 29
| 39 || 311 || [[2554edo|2554]]|| 311 || [[2398629edo|2398629]]|| 2398629
| 282
| 1323
|-
|-
| 31
| 41 || 311 || 2554 || 311 || [[19164767edo|19164767]]|| 19164767
| 311
| 1600
|-
|-
| 33
| 43 || [[17461edo|17461]]|| 17461 || [[20567edo|20567]]|| [[19735901edo|19735901]]|| 19735901
| 311
| 1600
|-
|-
| 35
| 45 || 17461 || 17461 || 20567 || 19735901 || 19735901
| 311
| 1600
|-
|-
| 37
| 47 || [[20567edo|20567]]|| 20567 || 20567 || [[152797015edo|152797015]]|| 152797015
| 311
| 1600
|-
|-
| 39
| 49 || 20567 || 20567 || [[459944edo|459944]]|| ||  
| 311
| 2554
|-
|-
| 41
| 51 || 20567 || 20567 || 459944 ||  ||  
| 311
| 2554
|-
|-
| 43
| 53 || 20567 || 20567 || [[1705229edo|1705229]]|| ||  
| 17461
| 17461
|-
|-
| 45
| 55 || 20567 || 20567 || 1705229 ||  ||  
| 17461
| 17461
|-
|-
| 47
| 57 || 20567 || 20567 || 1705229 ||  ||
| 20567
| 20567
|-
|-
| 49
| 59 || [[253389edo|253389]]|| 253389 || [[3159811edo|3159811]]|| ||  
| 20567
| 20567
|-
|-
| 51
| 61 || [[625534edo|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 || [[7317929edo|7317929]]|| ||  
| 20567
| 20567
|-
|-
| 57
| 69 || [[759630edo|759630]]|| 759630 || [[8595351edo|8595351]]|| ||  
| 20567
| 20567
|-
|-
| 59
| 71 || 759630 || 759630 || 8595351 ||  ||  
| 253389
| 253389
|-
|-
| 61
| 73 || 759630 || 759630 || [[27783092edo|27783092]]|| ||  
| 625534
| 625534
|-
|-
| 63
| 75 || [[2157429edo|2157429]]|| 2157429 || [[34531581edo|34531581]]|| ||  
| 625534
| 625534
|-
|-
| 65
| 77 || 2157429 || 2157429 || 34531581 ||  ||  
| 625534
| 625534
|-
|-
| 67
| 79 || [[2901533edo|2901533]]|| 2901533 || [[50203972edo|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 || [[1297643131edo|1297643131]]|| ||
| 2901533
| 2901533
|-
|-
| 87
| 99 || 2901533 || 2901533 || 1297643131 ||  ||
| 2901533
| 2901533
|-
|-
| 89
| 101 || 2901533 || 2901533 || [[3888109922edo|3888109922]]|| ||
| 2901533
| 2901533
|-
|-
| 91
| 103 || 2901533 || 2901533 || 3888109922 ||  ||
| 2901533
| 2901533
|-
|-
| 93
| 105 || 2901533 || 2901533 || 3888109922 ||  ||
| 2901533
| 2901533
|-
|-
| 95
| 107 || 2901533 || 2901533 || [[13805152233edo|13805152233]]|| ||
| 2901533
| 2901533
|-
|-
| 97
| 109 || 2901533 || 2901533 || [[27218556026edo|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 || [[42586208631edo|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 || [[93678217813edo|93678217813]]** || ||
| 2901533
| 2901533
|-
|-
| 119
| 133 || [[70910024edo|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
|-
| 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
|}
|}
<nowiki />* Apart from 0edo
<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 [[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}}
Line 327: Line 156:
* {{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}}


[[Category:Theory]]
== 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"