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. Odd limits of {{nowrap|2''n'' − 1}} are '''highlighted'''.

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''n'' − 1}} are '''highlighted'''.

{| class="wikitable center-all"

<onlyinclude>{| class="wikitable center-all"

! Odd limit !! Smallest consistent edo~~<ref group=note>apart from 0edo</ref>~~ !! Smallest distinctly consistent edo !! Smallest ''purely consistent''<~~ref group=note~~>~~''purely~~ consistent~~'' is an {{idiosyncratic}}~~<~~/ref~~> edo

|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit

|-

! Odd limit !! Smallest consistent edo* !! Smallest distinctly consistent edo !! Smallest purely consistent edo* !! Smallest edo consistent to [[Consistency #Generalization|distance 2]]* !! Smallest edo distinctly consistent to distance 2

|- style="font-weight: bold; background-color: #dddddd;"

| 1 || 1 || 1 || 1

| 1 || 1 || 1 || 1 || 1 || 1

|- style="font-weight: bold; background-color: #dddddd;"

| 3 || 1 || 3 || 2

| 3 || 1 || 3 || 2 || 2 || 3

|-

| 5 || 3 || 9 || 3

| 5 || 3 || 9 || 3 || 3 || 12

|- style="font-weight: bold; background-color: #dddddd;"

| 7 || 4 || 27 || 10

| 7 || 4 || 27 || 10 || 31 || 31

|-

| 9 || 5 || 41 || 41

| 9 || 5 || 41 || 41 || 41 || 41

|-

| 11 || 22 || 58 || 41

| 11 || 22 || 58 || 41 || 72 || 72

|-

| 13 || 26 || 87 || 46

| 13 || 26 || 87 || 46 || 270 || 270

|- style="font-weight: bold; background-color: #dddddd;"

| 15 || 29 || 111 || 87

| 15 || 29 || 111 || 87 || 494 || 494

|-

| 17 || 58 || 149 || 311

| 17 || 58 || 149 || 311 || 3395 || 3395

|-

| 19 || 80 || 217 || 311

| 19 || 80 || 217 || 311 || 8539 || 8539

|-

| 21 || 94 || 282 || 311

| 21 || 94 || 282 || 311 || 8539 || 8539

|-

| 23 || 94 || 282 || 311

| 23 || 94 || 282 || 311 || 16808 || 16808

|-

| 25 || 282 || 388 || 311

| 25 || 282 || 388 || 311 || 16808 || 16808

|-

| 27 || 282 || 388 || 311

| 27 || 282 || 388 || 311 || 16808 || 16808

|-

| 29 || 282 || 1323 || 311

| 29 || 282 || 1323 || 311 || 16808 || 16808

|- style="font-weight: bold; background-color: #dddddd;"

| 31 || 311 || 1600 || 311

| 31 || 311 || 1600 || 311 || 16808 || 16808

|-

| 33 || 311 || 1600 || 311

| 33 || 311 || 1600 || 311 || 16808 || 16808

|-

| 35 || 311 || 1600 || 311

| 35 || 311 || 1600 || 311 || 16808 || 16808

|-

| 37 || 311 || 1600 || 311

| 37 || 311 || 1600 || 311 || 324296 || 324296

|-

| 39 || 311 || 2554 || 311

| 39 || 311 || 2554 || 311 || 2398629 || 2398629

|-

| 41 || 311 || 2554 || 311

| 41 || 311 || 2554 || 311 || 19164767 || 19164767

|-

| 43 || 17461 || 17461 || 20567

| 43 || 17461 || 17461 || 20567 || 19735901 || 19735901

|-

| 45 || 17461 || 17461 || 20567

| 45 || 17461 || 17461 || 20567 || 19735901 || 19735901

|-

| 47 || 20567 || 20567 || 20567

| 47 || 20567 || 20567 || 20567 || 152797015 || 152797015

|-

| 49 || 20567 || 20567 || 459944

| 49 || 20567 || 20567 || 459944 || ||

|-

| 51 || 20567 || 20567 || 459944

| 51 || 20567 || 20567 || 459944 || ||

|-

| 53 || 20567 || 20567 || 1705229

| 53 || 20567 || 20567 || 1705229 || ||

|-

| 55 || 20567 || 20567 || 1705229

| 55 || 20567 || 20567 || 1705229 || ||

|-

| 57 || 20567 || 20567 || 1705229

| 57 || 20567 || 20567 || 1705229 || ||

|-

| 59 || 253389 || 253389 || 3159811

| 59 || 253389 || 253389 || 3159811 || ||

|-

| 61 || 625534 || 625534 || 3159811

| 61 || 625534 || 625534 || 3159811 || ||

|- style="font-weight: bold; background-color: #dddddd;"

| 63 || 625534 || 625534 || 3159811

| 63 || 625534 || 625534 || 3159811 || ||

|-

| 65 || 625534 || 625534 || 3159811

| 65 || 625534 || 625534 || 3159811 || ||

|-

| 67 || 625534 || 625534 || 7317929

| 67 || 625534 || 625534 || 7317929 || ||

|-

| 69 || 759630 || 759630 || 8595351

| 69 || 759630 || 759630 || 8595351 || ||

|-

| 71 || 759630 || 759630 || 8595351

| 71 || 759630 || 759630 || 8595351 || ||

|-

| 73 || 759630 || 759630 || 27783092

| 73 || 759630 || 759630 || 27783092 || ||

|-

| 75 || 2157429 || 2157429 || 34531581

| 75 || 2157429 || 2157429 || 34531581 || ||

|-

| 77 || 2157429 || 2157429 || 34531581

| 77 || 2157429 || 2157429 || 34531581 || ||

|-

| 79 || 2901533 || 2901533 || 50203972

| 79 || 2901533 || 2901533 || 50203972 || ||

|-

| 81 || 2901533 || 2901533 || 50203972

| 81 || 2901533 || 2901533 || 50203972 || ||

|-

| 83 || 2901533 || 2901533 || 50203972

| 83 || 2901533 || 2901533 || 50203972 || ||

|-

| 85 || 2901533 || 2901533 || 50203972

| 85 || 2901533 || 2901533 || 50203972 || ||

|-

| 87 || 2901533 || 2901533 || 50203972

| 87 || 2901533 || 2901533 || 50203972 || ||

|-

| 89 || 2901533 || 2901533 || 50203972

| 89 || 2901533 || 2901533 || 50203972 || ||

|-

| 91 || 2901533 || 2901533 || 50203972

| 91 || 2901533 || 2901533 || 50203972 || ||

|-

| 93 || 2901533 || 2901533 || 50203972

| 93 || 2901533 || 2901533 || 50203972 || ||

|-

| 95 || 2901533 || 2901533 || 50203972

| 95 || 2901533 || 2901533 || 50203972 || ||

|-

| 97 || 2901533 || 2901533 || 1297643131

| 97 || 2901533 || 2901533 || 1297643131 || ||

|-

| 99 || 2901533 || 2901533 || 1297643131

| 99 || 2901533 || 2901533 || 1297643131 || ||

|-

| 101 || 2901533 || 2901533 || 3888109922

| 101 || 2901533 || 2901533 || 3888109922 || ||

|-

| 103 || 2901533 || 2901533 || 3888109922

| 103 || 2901533 || 2901533 || 3888109922 || ||

|-

| 105 || 2901533 || 2901533 || 3888109922

| 105 || 2901533 || 2901533 || 3888109922 || ||

|-

| 107 || 2901533 || 2901533 || 13805152233

| 107 || 2901533 || 2901533 || 13805152233 || ||

|-

| 109 || 2901533 || 2901533 || 27218556026

| 109 || 2901533 || 2901533 || 27218556026 || ||

|-

| 111 || 2901533 || 2901533 || 27218556026

| 111 || 2901533 || 2901533 || 27218556026 || ||

|-

| 113 || 2901533 || 2901533 || 27218556026

| 113 || 2901533 || 2901533 || 27218556026 || ||

|-

| 115 || 2901533 || 2901533 || 27218556026

| 115 || 2901533 || 2901533 || 27218556026 || ||

|-

| 117 || 2901533 || 2901533 || 27218556026

| 117 || 2901533 || 2901533 || 27218556026 || ||

|-

| 119 || 2901533 || 2901533 || 42586208631

| 119 || 2901533 || 2901533 || 42586208631 || ||

|-

| 121 || 2901533 || 2901533 || 42586208631

| 121 || 2901533 || 2901533 || 42586208631 || ||

|-

| 123 || 2901533 || 2901533 || 42586208631

| 123 || 2901533 || 2901533 || 42586208631 || ||

|-

| 125 || 2901533 || 2901533 || 42586208631

| 125 || 2901533 || 2901533 || 42586208631 || ||

|- style="font-weight: bold; background-color: #dddddd;"

| 127 || 2901533 || 2901533 || 42586208631

| 127 || 2901533 || 2901533 || 42586208631 || ||

|-

| 129 || 2901533 || 2901533 || 42586208631

| 129 || 2901533 || 2901533 || 42586208631 || ||

|-

| 131 || 2901533 || 2901533 || 93678217813

| 131 || 2901533 || 2901533 || 93678217813** || ||

|-

| 133 || 70910024 || 70910024 || 93678217813

| 133 || 70910024 || 70910024 || 93678217813 || ||

|-

| 135 || 70910024 || 70910024 || 93678217813~~<ref group=note>purely consistent to the 137-odd-limit</ref>~~

| 135 || 70910024 || 70910024 || 93678217813 || ||

|}

<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.

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== See also ==

* [[Consistency limits of small EDOs]]

* {{u|ArrowHead294|Purely consistent EDOs by odd limit}}

[[Category:Mapping]]

[[Category:Consistency]]

[[Category:Odd limit]]

~~== Notes ==~~

~~<references group="note"/>~~

@@ Line 1: / Line 1: @@
-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'''.
+{{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 center-all"
+<onlyinclude>{| class="wikitable center-all"
-! Odd<br />limit !! Smallest<br />consistent edo<ref group=note>apart from 0edo</ref> !! Smallest distinctly<br />consistent edo !! Smallest ''purely<br />consistent''<ref group=note>''purely consistent'' is an {{idiosyncratic}}</ref> edo
+|+ style="font-size: 105%;" | Smallest consistent EDOs per 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
 |- style="font-weight: bold; background-color: #dddddd;"
-| 1 || 1 || 1 || 1
+| 1 || 1 || 1 || 1 || 1 || 1
 |- style="font-weight: bold; background-color: #dddddd;"
-| 3 || 1 || 3 || 2
+| 3 || 1 || 3 || 2 || 2 || 3
 |-
-| 5 || 3 || 9 || 3
+| 5 || 3 || 9 || 3 || 3 || 12
 |- style="font-weight: bold; background-color: #dddddd;"
-| 7 || 4 || 27 || 10
+| 7 || 4 || 27 || 10 || 31 || 31
 |-
-| 9 || 5 || 41 || 41
+| 9 || 5 || 41 || 41 || 41 || 41
 |-
-| 11 || 22 || 58 || 41
+| 11 || 22 || 58 || 41 || 72 || 72
 |-
-| 13 || 26 || 87 || 46
+| 13 || 26 || 87 || 46 || 270 || 270
 |- style="font-weight: bold; background-color: #dddddd;"
-| 15 || 29 || 111 || 87
+| 15 || 29 || 111 || 87 || 494 || 494
 |-
-| 17 || 58 || 149 || 311
+| 17 || 58 || 149 || 311 || 3395 || 3395
 |-
-| 19 || 80 || 217 || 311
+| 19 || 80 || 217 || 311 || 8539 || 8539
 |-
-| 21 || 94 || 282 || 311
+| 21 || 94 || 282 || 311 || 8539 || 8539
 |-
-| 23 || 94 || 282 || 311
+| 23 || 94 || 282 || 311 || 16808 || 16808
 |-
-| 25 || 282 || 388 || 311
+| 25 || 282 || 388 || 311 || 16808 || 16808
 |-
-| 27 || 282 || 388 || 311
+| 27 || 282 || 388 || 311 || 16808 || 16808
 |-
-| 29 || 282 || 1323 || 311
+| 29 || 282 || 1323 || 311 || 16808 || 16808
 |- style="font-weight: bold; background-color: #dddddd;"
-| 31 || 311 || 1600 || 311
+| 31 || 311 || 1600 || 311 || 16808 || 16808
 |-
-| 33 || 311 || 1600 || 311
+| 33 || 311 || 1600 || 311 || 16808 || 16808
 |-
-| 35 || 311 || 1600 || 311
+| 35 || 311 || 1600 || 311 || 16808 || 16808
 |-
-| 37 || 311 || 1600 || 311
+| 37 || 311 || 1600 || 311 || 324296 || 324296
 |-
-| 39 || 311 || 2554 || 311
+| 39 || 311 || 2554 || 311 || 2398629 || 2398629
 |-
-| 41 || 311 || 2554 || 311
+| 41 || 311 || 2554 || 311 || 19164767 || 19164767
 |-
-| 43 || 17461 || 17461 || 20567
+| 43 || 17461 || 17461 || 20567 || 19735901 || 19735901
 |-
-| 45 || 17461 || 17461 || 20567
+| 45 || 17461 || 17461 || 20567 || 19735901 || 19735901
 |-
-| 47 || 20567 || 20567 || 20567
+| 47 || 20567 || 20567 || 20567 || 152797015 || 152797015
 |-
-| 49 || 20567 || 20567 || 459944
+| 49 || 20567 || 20567 || 459944 ||  ||
 |-
-| 51 || 20567 || 20567 || 459944
+| 51 || 20567 || 20567 || 459944 ||  ||
 |-
-| 53 || 20567 || 20567 || 1705229
+| 53 || 20567 || 20567 || 1705229 ||  ||
 |-
-| 55 || 20567 || 20567 || 1705229
+| 55 || 20567 || 20567 || 1705229 ||  ||
 |-
-| 57 || 20567 || 20567 || 1705229
+| 57 || 20567 || 20567 || 1705229 ||  ||
 |-
-| 59 || 253389 || 253389 || 3159811
+| 59 || 253389 || 253389 || 3159811 ||  ||
 |-
-| 61 || 625534 || 625534 || 3159811
+| 61 || 625534 || 625534 || 3159811 ||  ||
 |- style="font-weight: bold; background-color: #dddddd;"
-| 63 || 625534 || 625534 || 3159811
+| 63 || 625534 || 625534 || 3159811 ||  ||
 |-
-| 65 || 625534 || 625534 || 3159811
+| 65 || 625534 || 625534 || 3159811 ||  ||
 |-
-| 67 || 625534 || 625534 || 7317929
+| 67 || 625534 || 625534 || 7317929 ||  ||
 |-
-| 69 || 759630 || 759630 || 8595351
+| 69 || 759630 || 759630 || 8595351 ||  ||
 |-
-| 71 || 759630 || 759630 || 8595351
+| 71 || 759630 || 759630 || 8595351 ||  ||
 |-
-| 73 || 759630 || 759630 || 27783092
+| 73 || 759630 || 759630 || 27783092 ||  ||
 |-
-| 75 || 2157429 || 2157429 || 34531581
+| 75 || 2157429 || 2157429 || 34531581 ||  ||
 |-
-| 77 || 2157429 || 2157429 || 34531581
+| 77 || 2157429 || 2157429 || 34531581 ||  ||
 |-
-| 79 || 2901533 || 2901533 || 50203972
+| 79 || 2901533 || 2901533 || 50203972 ||  ||
 |-
-| 81 || 2901533 || 2901533 || 50203972
+| 81 || 2901533 || 2901533 || 50203972 ||  ||
 |-
-| 83 || 2901533 || 2901533 || 50203972
+| 83 || 2901533 || 2901533 || 50203972 ||  ||
 |-
-| 85 || 2901533 || 2901533 || 50203972
+| 85 || 2901533 || 2901533 || 50203972 ||  ||
 |-
-| 87 || 2901533 || 2901533 || 50203972
+| 87 || 2901533 || 2901533 || 50203972 ||  ||
 |-
-| 89 || 2901533 || 2901533 || 50203972
+| 89 || 2901533 || 2901533 || 50203972 ||  ||
 |-
-| 91 || 2901533 || 2901533 || 50203972
+| 91 || 2901533 || 2901533 || 50203972 ||  ||
 |-
-| 93 || 2901533 || 2901533 || 50203972
+| 93 || 2901533 || 2901533 || 50203972 ||  ||
 |-
-| 95 || 2901533 || 2901533 || 50203972
+| 95 || 2901533 || 2901533 || 50203972 ||  ||
 |-
-| 97 || 2901533 || 2901533 || 1297643131
+| 97 || 2901533 || 2901533 || 1297643131 ||  ||
 |-
-| 99 || 2901533 || 2901533 || 1297643131
+| 99 || 2901533 || 2901533 || 1297643131 ||  ||
 |-
-| 101 || 2901533 || 2901533 || 3888109922
+| 101 || 2901533 || 2901533 || 3888109922 ||  ||
 |-
-| 103 || 2901533 || 2901533 || 3888109922
+| 103 || 2901533 || 2901533 || 3888109922 ||  ||
 |-
-| 105 || 2901533 || 2901533 || 3888109922
+| 105 || 2901533 || 2901533 || 3888109922 ||  ||
 |-
-| 107 || 2901533 || 2901533 || 13805152233
+| 107 || 2901533 || 2901533 || 13805152233 ||  ||
 |-
-| 109 || 2901533 || 2901533 || 27218556026
+| 109 || 2901533 || 2901533 || 27218556026 ||  ||
 |-
-| 111 || 2901533 || 2901533 || 27218556026
+| 111 || 2901533 || 2901533 || 27218556026 ||  ||
 |-
-| 113 || 2901533 || 2901533 || 27218556026
+| 113 || 2901533 || 2901533 || 27218556026 ||  ||
 |-
-| 115 || 2901533 || 2901533 || 27218556026
+| 115 || 2901533 || 2901533 || 27218556026 ||  ||
 |-
-| 117 || 2901533 || 2901533 || 27218556026
+| 117 || 2901533 || 2901533 || 27218556026 ||  ||
 |-
-| 119 || 2901533 || 2901533 || 42586208631
+| 119 || 2901533 || 2901533 || 42586208631 ||  ||
 |-
-| 121 || 2901533 || 2901533 || 42586208631
+| 121 || 2901533 || 2901533 || 42586208631 ||  ||
 |-
-| 123 || 2901533 || 2901533 || 42586208631
+| 123 || 2901533 || 2901533 || 42586208631 ||  ||
 |-
-| 125 || 2901533 || 2901533 || 42586208631
+| 125 || 2901533 || 2901533 || 42586208631 ||  ||
 |- style="font-weight: bold; background-color: #dddddd;"
-| 127 || 2901533 || 2901533 || 42586208631
+| 127 || 2901533 || 2901533 || 42586208631 ||  ||
 |-
-| 129 || 2901533 || 2901533 || 42586208631
+| 129 || 2901533 || 2901533 || 42586208631 ||  ||
 |-
-| 131 || 2901533 || 2901533 || 93678217813
+| 131 || 2901533 || 2901533 || 93678217813** ||  ||
 |-
-| 133 || 70910024 || 70910024 || 93678217813
+| 133 || 70910024 || 70910024 || 93678217813 ||  ||
 |-
-| 135 || 70910024 || 70910024 || 93678217813<ref group=note>purely consistent to the 137-odd-limit</ref>
+| 135 || 70910024 || 70910024 || 93678217813 ||  ||
 |}
+<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.
@@ Line 151: / Line 157: @@
 == See also ==
 * [[Consistency limits of small EDOs]]
+* {{u|ArrowHead294|Purely consistent EDOs by odd limit}}
 [[Category:Mapping]]
 [[Category:Consistency]]
 [[Category:Odd limit]]
-== Notes ==
-<references group="note"/>