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
| 3
|-
|-
| 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
| 459944
|-
|-
| 51
| 61 || 625534 || 625534 || 3159811 ||  ||
| 20567
|- style="font-weight: bold; background-color: #dddddd;"
| 20567
| 63 || 625534 || 625534 || 3159811 ||  ||  
| 459944
|-
|-
| 53
| 65 || 625534 || 625534 || 3159811 ||  ||  
| 20567
| 20567
| 1705229
|-
|-
| 55
| 67 || 625534 || 625534 || 7317929 ||  ||  
| 20567
| 20567
| 1705229
|-
|-
| 57
| 69 || 759630 || 759630 || 8595351 ||  ||  
| 20567
| 20567
| 1705229
|-
|-
| 59
| 71 || 759630 || 759630 || 8595351 ||  ||  
| 253389
| 253389
| 3159811
|-
|-
| 61
| 73 || 759630 || 759630 || 27783092 ||  ||  
| 625534
| 625534
| 3159811
|-
|-
| 63
| 75 || 2157429 || 2157429 || 34531581 ||  ||  
| 625534
| 625534
| 3159811
|-
|-
| 65
| 77 || 2157429 || 2157429 || 34531581 ||  ||  
| 625534
| 625534
| 3159811
|-
|-
| 67
| 79 || 2901533 || 2901533 || 50203972 ||  ||  
| 625534
| 625534
| 7317929
|-
|-
| 69
| 81 || 2901533 || 2901533 || 50203972 ||  ||  
| 759630
| 759630
| 8595351
|-
|-
| 71
| 83 || 2901533 || 2901533 || 50203972 ||  ||  
| 759630
| 759630
| 8595351
|-
|-
| 73
| 85 || 2901533 || 2901533 || 50203972 ||  ||  
| 759630
| 759630
| 27783092
|-
|-
| 75
| 87 || 2901533 || 2901533 || 50203972 ||  ||  
| 2157429
| 2157429
| 34531581
|-
|-
| 77
| 89 || 2901533 || 2901533 || 50203972 ||  ||  
| 2157429
| 2157429
| 34531581
|-
|-
| 79
| 91 || 2901533 || 2901533 || 50203972 ||  ||
| 2901533
| 2901533
| 50203972
|-
|-
| 81
| 93 || 2901533 || 2901533 || 50203972 ||  ||
| 2901533
| 2901533
| 50203972
|-
|-
| 83
| 95 || 2901533 || 2901533 || 50203972 ||  ||
| 2901533
| 2901533
| 50203972
|-
|-
| 85
| 97 || 2901533 || 2901533 || 1297643131 ||  ||
| 2901533
| 2901533
| 50203972
|-
|-
| 87
| 99 || 2901533 || 2901533 || 1297643131 ||  ||
| 2901533
| 2901533
| 50203972
|-
|-
| 89
| 101 || 2901533 || 2901533 || 3888109922 ||  ||
| 2901533
| 2901533
| 50203972
|-
|-
| 91
| 103 || 2901533 || 2901533 || 3888109922 ||  ||
| 2901533
| 2901533
| 50203972
|-
|-
| 93
| 105 || 2901533 || 2901533 || 3888109922 ||  ||
| 2901533
| 2901533
| 50203972
|-
|-
| 95
| 107 || 2901533 || 2901533 || 13805152233 ||  ||
| 2901533
| 2901533
| 50203972
|-
|-
| 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 [[5407372813edo|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.
Line 359: Line 154:
* {{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]]

Latest revision as of 19:06, 14 August 2025

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

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, and purely consistent[idiosyncratic term] if its 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 2n − 1 are highlighted.

Smallest consistent EDOs per odd limit
Odd
limit		 Smallest
consistent edo*		 Smallest distinctly
consistent edo		 Smallest purely
consistent edo*		 Smallest edo
consistent to
distance 2*		 Smallest edo
distinctly consistent
to distance 2
1 1 1 1 1 1
3 1 3 2 2 3
5 3 9 3 3 12
7 4 27 10 31 31
9 5 41 41 41 41
11 22 58 41 72 72
13 26 87 46 270 270
15 29 111 87 494 494
17 58 149 311 3395 3395
19 80 217 311 8539 8539
21 94 282 311 8539 8539
23 94 282 311 16808 16808
25 282 388 311 16808 16808
27 282 388 311 16808 16808
29 282 1323 311 16808 16808
31 311 1600 311 16808 16808
33 311 1600 311 16808 16808
35 311 1600 311 16808 16808
37 311 1600 311 324296 324296
39 311 2554 311 2398629 2398629
41 311 2554 311 19164767 19164767
43 17461 17461 20567 19735901 19735901
45 17461 17461 20567 19735901 19735901
47 20567 20567 20567 152797015 152797015
49 20567 20567 459944
51 20567 20567 459944
53 20567 20567 1705229
55 20567 20567 1705229
57 20567 20567 1705229
59 253389 253389 3159811
61 625534 625534 3159811
63 625534 625534 3159811
65 625534 625534 3159811
67 625534 625534 7317929
69 759630 759630 8595351
71 759630 759630 8595351
73 759630 759630 27783092
75 2157429 2157429 34531581
77 2157429 2157429 34531581
79 2901533 2901533 50203972
81 2901533 2901533 50203972
83 2901533 2901533 50203972
85 2901533 2901533 50203972
87 2901533 2901533 50203972
89 2901533 2901533 50203972
91 2901533 2901533 50203972
93 2901533 2901533 50203972
95 2901533 2901533 50203972
97 2901533 2901533 1297643131
99 2901533 2901533 1297643131
101 2901533 2901533 3888109922
103 2901533 2901533 3888109922
105 2901533 2901533 3888109922
107 2901533 2901533 13805152233
109 2901533 2901533 27218556026
111 2901533 2901533 27218556026
113 2901533 2901533 27218556026
115 2901533 2901533 27218556026
117 2901533 2901533 27218556026
119 2901533 2901533 42586208631
121 2901533 2901533 42586208631
123 2901533 2901533 42586208631
125 2901533 2901533 42586208631
127 2901533 2901533 42586208631
129 2901533 2901533 42586208631
131 2901533 2901533 93678217813**
133 70910024 70910024 93678217813
135 70910024 70910024 93678217813

* Apart from 0edo

** Purely consistent to the 137-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.

OEIS integer sequences links

See also

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