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<sup>''n''</sup> &minus; 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'' if its [[relative interval error|relative errors]] on odd harmonics up to and including ''q'' never exceed 25%. It is accurate if every interval in that odd-limit are consistent to at least [[Harmonic distance|distance 2]], or at most 25% relative error. 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'''.  


<onlyinclude>{| class="wikitable center-all"
<onlyinclude><includeonly>
<includeonly>
|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit
|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit
</includeonly>
</includeonly>
{| class="wikitable center-all"
|-
|-
! Odd<br />limit !! Smallest<br />consistent edo&#42; !! Smallest distinctly<br />consistent edo !! Smallest ''purely<br />consistent''&#42;&#42; edo
! Odd<br />limit !! Smallest<br />consistent edo&#42; !! Smallest distinctly<br />consistent edo !! Smallest ''purely<br />consistent''&#42;&#42; edo
!Smallest
accurate**** edo
!Smallest distinctly
accurate**** edo
|- style="font-weight: bold; background-color: #dddddd;"
|- style="font-weight: bold; background-color: #dddddd;"
| 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1
|1
|1
|- style="font-weight: bold; background-color: #dddddd;"
|- 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;"
|- 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;"
|- 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
|
|
|-
|-
| 19 || 80 || 217 || 311
| 19 || 80 || 217 || 311
|
|
|-
|-
| 21 || 94 || 282 || 311
| 21 || 94 || 282 || 311
|
|
|-
|-
| 23 || 94 || 282 || 311
| 23 || 94 || 282 || 311
|
|
|-
|-
| 25 || 282 || 388 || 311
| 25 || 282 || 388 || 311
|
|
|-
|-
| 27 || 282 || 388 || 311
| 27 || 282 || 388 || 311
|
|
|-
|-
| 29 || 282 || 1323 || 311
| 29 || 282 || 1323 || 311
|
|
|- style="font-weight: bold; background-color: #dddddd;"
|- style="font-weight: bold; background-color: #dddddd;"
| 31 || 311 || 1600 || 311
| 31 || 311 || 1600 || 311
|
|
|-
|-
| 33 || 311 || 1600 || 311
| 33 || 311 || 1600 || 311
|
|
|-
|-
| 35 || 311 || 1600 || 311
| 35 || 311 || 1600 || 311
|
|
|-
|-
| 37 || 311 || 1600 || 311
| 37 || 311 || 1600 || 311
|
|
|-
|-
| 39 || 311 || 2554 || 311
| 39 || 311 || 2554 || 311
|
|
|-
|-
| 41 || 311 || 2554 || 311
| 41 || 311 || 2554 || 311
|
|
|-
|-
| 43 || 17461 || 17461 || 20567
| 43 || 17461 || 17461 || 20567
|
|
|-
|-
| 45 || 17461 || 17461 || 20567
| 45 || 17461 || 17461 || 20567
|
|
|-
|-
| 47 || 20567 || 20567 || 20567
| 47 || 20567 || 20567 || 20567
|
|
|-
|-
| 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;"
|- 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;"
|- 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
| 135 || 70910024 || 70910024 || 93678217813
{{Table notes|cols=4
 
|
|{{Table notes|cols=4
| Apart from 0edo
| Apart from 0edo
| ''Purely consistent'' is an {{idiosyncratic}}
| ''Purely consistent'' is an {{idiosyncratic}}
| Purely consistent to the 137-odd-limit
| Purely consistent to the 137-odd-limit
}}
}}
|}</onlyinclude>
|}
</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.

Revision as of 03:00, 10 September 2024

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 if its relative errors on odd harmonics up to and including q never exceed 25%. It is accurate if every interval in that odd-limit are consistent to at least distance 2, or at most 25% relative error. 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.


Odd
limit		 Smallest
consistent edo*		 Smallest distinctly
consistent edo		 Smallest purely
consistent** edo 		Smallest

accurate**** edo

Smallest distinctly

accurate**** edo

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
19 80 217 311
21 94 282 311
23 94 282 311
25 282 388 311
27 282 388 311
29 282 1323 311
31 311 1600 311
33 311 1600 311
35 311 1600 311
37 311 1600 311
39 311 2554 311
41 311 2554 311
43 17461 17461 20567
45 17461 17461 20567
47 20567 20567 20567
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 Template:Table notes


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

Retrieved from "https://en.xen.wiki/index.php?title=Minimal_consistent_EDOs&oldid=154497"