Xenharmonic Wiki

Minimal consistent EDOs: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
4,516 edits
mNo edit summary
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
 
(64 intermediate revisions by 10 users not shown)
Line 1: Line 1:
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.
{{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"
<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
|- style="font-weight: bold; background-color: #dddddd;"
! | Smallest uniquely consistent
| 1 || 1 || 1 || 1 || 1 || 1
|- 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
|-
|-
| | 3
| 9 || 5 || 41 || 41 || 41 || 41
| | 1
| | 3
|-
|-
| | 5
| 11 || 22 || 58 || 41 || 72 || 72
| | 3
| | 9
|-
|-
| | 7
| 13 || 26 || 87 || 46 || 270 || 270
| | 4
|- style="font-weight: bold; background-color: #dddddd;"
| | 27
| 15 || 29 || 111 || 87 || 494 || 494
|-
|-
| | 9
| 17 || 58 || 149 || 311 || 3395 || 3395
| | 5
| | 41
|-
|-
| | 11
| 19 || 80 || 217 || 311 || 8539 || 8539
| | 22
| | 58
|-
|-
| | 13
| 21 || 94 || 282 || 311 || 8539 || 8539
| | 26
| | 87
|-
|-
| | 15
| 23 || 94 || 282 || 311 || 16808 || 16808
| | 29
| | 111
|-
|-
| | 17
| 25 || 282 || 388 || 311 || 16808 || 16808
| | 58
| | 149
|-
|-
| | 19
| 27 || 282 || 388 || 311 || 16808 || 16808
| | 80
| | 217
|-
|-
| | 21
| 29 || 282 || 1323 || 311 || 16808 || 16808
| | 94
|- style="font-weight: bold; background-color: #dddddd;"
| | 282
| 31 || 311 || 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 || 324296 || 324296
| | 282
| | 388
|-
|-
| | 29
| 39 || 311 || 2554 || 311 || 2398629 || 2398629
| | 282
| | 1323
|-
|-
| | 31
| 41 || 311 || 2554 || 311 || 19164767 || 19164767
| | 311
| | 1600
|-
|-
| | 33
| 43 || 17461 || 17461 || 20567 || 19735901 || 19735901
| | 311
| | 1600
|-
|-
| | 35
| 45 || 17461 || 17461 || 20567 || 19735901 || 19735901
| | 311
| | 1600
|-
|-
| | 37
| 47 || 20567 || 20567 || 20567 || 152797015 || 152797015
| | 311
| | 1600
|-
|-
| | 39
| 49 || 20567 || 20567 || 459944 ||  ||  
| | 311
| | 2554
|-
|-
| | 41
| 51 || 20567 || 20567 || 459944 ||  ||  
| | 311
| | 2554
|-
|-
| | 43
| 53 || 20567 || 20567 || 1705229 ||  ||  
| | 17461
| | 17461
|-
|-
| | 45
| 55 || 20567 || 20567 || 1705229 ||  ||  
| | 17461
| | 17461
|-
|-
| | 47
| 57 || 20567 || 20567 || 1705229 ||  ||
| | 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
|-
|-
| | 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
|-
| | 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


=OEIS integer sequences links=
<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|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:Consistency]]
[[Category:Odd limit]]
Retrieved from "https://en.xen.wiki/w/Minimal_consistent_EDOs"