Consistency limits of small EDOs: Difference between revisions

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An [[EDO|edo]] N is [[consistent|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 every edo up to 99. "Consistent" gives the consistency level, and "Distinct" the distinct consistency level.
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. Below is a table of every edo up to 99. "Consistent" gives its consistency limit, i.e. the highest odd limit to which the edo is consistent, and "Distinct" gives the distinct consistency limit, i.e. the highest odd limit to which the edo is distinctly consistent. The remaining columns give the [[Consistent #Consistency to distance d|consistency distance]] (also called ''consistency level''<ref group="note">This term was coined by [[Paul Hahn]] in 1996. See [https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html Yahoo! Tuning Group | ''Consistency generalized'']. </ref>) for every odd limit from 3 to 23.


{| class="wikitable"
{| class="wikitable sortable mw-collapsible right-all"
|-
|-
| | EDO
! rowspan="2" | Edo
| | Consistent
! colspan="2" | Consistency limit
| | Distinct
! colspan="11" | Consistency distance
|-
|-
| | 1
! Consistent
| | 3
! Distinct
| | 1
! 3-limit
! 5-limit
! 7-limit
! 9-limit
! 11-limit
! 13-limit
! 15-limit
! 17-limit
! 19-limit
! 21-limit
! 23-limit
|-
|-
| | 2
| 1 || style="background-color: #fefefe;"| 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 1
|-
|-
| | 3
| 2 || style="background-color: #fefefe;"| 3 || 1 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 3
|-
|-
| | 4
| 3 || style="background-color: #fefefe;"| 5 || 3 || 2 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 1
|-
|-
| | 5
| 4 || style="background-color: #fefefe;"| 7 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 3
|-
|-
| | 6
| 5 || style="background-color: #fefefe;"| 9 || 3 || 6 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 3
|-
|-
| | 7
| 6 || style="background-color: #fefefe;"| 7 || 3 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 3
|-
|-
| | 8
| 7 || style="background-color: #fefefe;"| 5 || 3 || 5 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 3
|-
|-
| | 9
| 8 || style="background-color: #fefefe;"| 5 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 5
|-
|-
| | 10
| 9 || style="background-color: #fefefe;"| 7 || 5 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 3
|-
|-
| | 11
| 10 || style="background-color: #fefefe;"| 7 || 3 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 12
| 11 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 5
|-
|-
| | 13
| 12 || style="background-color: #fefefe;"| 9 || 5 || 25 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 14
| 13 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 15
| 14 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 5
|-
|-
| | 16
| 15 || style="background-color: #fefefe;"| 7 || 5 || 2 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 5
|-
|-
| | 17
| 16 || style="background-color: #fefefe;"| 7 || 5 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 18
| 17 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 8 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 5
|-
|-
| | 19
| 18 || style="background-color: #fefefe;"| 7 || 5 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 5
|-
|-
| | 20
| 19 || style="background-color: #fefefe;"| 9 || 5 || 4 || 4 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 21
| 20 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 22
| 21 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 11
| | 5
|-
|-
| | 23
| 22 || style="background-color: #fefefe;"| 11 || 5 || 3 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 24
| 23 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 25
| 24 || style="background-color: #fefefe;"| 5 || 5 || 12 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 26
| 25 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 13
| | 5
|-
|-
| | 27
| 26 || style="background-color: #fefefe;"| 13 || 5 || 2 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0
| | 9
| | 7
|-
|-
| | 28
| 27 || style="background-color: #fefefe;"| 9 || 7 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 29
| 28 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 15
| | 5
|-
|-
| | 30
| 29 || style="background-color: #fefefe;"| 15 || 5 || 13 || 1 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 31
| 30 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 11
| | 7
|-
|-
| | 32
| 31 || style="background-color: #fefefe;"| 11 || 7 || 3 || 3 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 33
| 32 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 34
| 33 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 35
| 34 || style="background-color: #fefefe;"| 5 || 5 || 4 || style="background-color: #fefefe;"| 4 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 36
| 35 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 37
| 36 || style="background-color: #fefefe;"| 7 || 7 || 8 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 38
| 37 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 39
| 38 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 40
| 39 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 41
| 40 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 15
| | 9
|-
|-
| | 42
| 41 || style="background-color: #fefefe;"| 15 || 9 || 30 || 2 || 2 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 43
| 42 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 44
| 43 || style="background-color: #fefefe;"| 7 || 7 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 45
| 44 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 46
| 45 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 13
| | 9
|-
|-
| | 47
| 46 || style="background-color: #fefefe;"| 13 || 9 || 5 || 2 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 48
| 47 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 49
| 48 || style="background-color: #fefefe;"| 5 || 5 || 6 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 50
| 49 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 7
|-
|-
| | 51
| 50 || style="background-color: #fefefe;"| 9 || 7 || 2 || 2 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 52
| 51 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 53
| 52 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 9
|-
|-
| | 54
| 53 || style="background-color: #fefefe;"| 9 || 9 || 165 || 8 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 55
| 54 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 56
| 55 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 57
| 56 || style="background-color: #fefefe;"| 7 || 7 || 2 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 58
| 57 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 17
| | 11
|-
|-
| | 59
| 58 || style="background-color: #fefefe;"| 17 || 11 || 6 || 1 || 1 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 60
| 59 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 9
|-
|-
| | 61
| 60 || style="background-color: #fefefe;"| 9 || 9 || 5 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 62
| 61 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 63
| 62 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 64
| 63 || style="background-color: #fefefe;"| 7 || 7 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 65
| 64 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 66
| 65 || style="background-color: #fefefe;"| 5 || 5 || 22 || style="background-color: #fefefe;"| 5 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 67
| 66 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 68
| 67 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 9
|-
|-
| | 69
| 68 || style="background-color: #fefefe;"| 9 || 9 || 2 || 2 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 70
| 69 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 9
|-
|-
| | 71
| 70 || style="background-color: #fefefe;"| 9 || 9 || 9 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 72
| 71 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 17
| | 11
|-
|-
| | 73
| 72 || style="background-color: #fefefe;"| 17 || 11 || 4 || 2 || 2 || 2 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 74
| 73 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 75
| 74 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 76
| 75 || style="background-color: #fefefe;"| 5 || 5 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 77
| 76 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 9
|-
|-
| | 78
| 77 || style="background-color: #fefefe;"| 9 || 9 || 11 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 79
| 78 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 80
| 79 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 19
| | 11
|-
|-
| | 81
| 80 || style="background-color: #fefefe;"| 19 || 11 || 2 || 2 || 1 || 1 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0
| | 7
| | 7
|-
|-
| | 82
| 81 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 9
|-
|-
| | 83
| 82 || style="background-color: #fefefe;"| 9 || 9 || 15 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 84
| 83 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 9
|-
|-
| | 85
| 84 || style="background-color: #fefefe;"| 9 || 9 || 3 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 86
| 85 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 87
| 86 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 15
| | 13
|-
|-
| | 88
| 87 || style="background-color: #fefefe;"| 15 || 13 || 4 || 4 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 89
| 88 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 11
| | 11
|-
|-
| | 90
| 89 || style="background-color: #fefefe;"| 11 || 11 || 8 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 91
| 90 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
| | 9
|-
|-
| | 92
| 91 || style="background-color: #fefefe;"| 9 || 9 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 93
| 92 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 7
| | 7
|-
|-
| | 94
| 93 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 23
| | 13
|-
|-
| | 95
| 94 || style="background-color: #fefefe;"| 23 || 13 || 36 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1
| | 7
| | 7
|-
|-
| | 96
| 95 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 97
| 96 || style="background-color: #fefefe;"| 5 || 5 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 5
| | 5
|-
|-
| | 98
| 97 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 3
| | 3
|-
|-
| | 99
| 98 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| | 9
|-
| | 9
| 99 || style="background-color: #fefefe;"| 9 || 9 || 5 || 3 || 3 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|}
|}


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


[[Category:Theory]]
[[Category:Mapping]]
[[Category:Mapping]]
[[Category:Consistency]]
[[Category:Odd limit]]

Latest revision as of 11:49, 7 August 2025

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. Below is a table of every edo up to 99. "Consistent" gives its consistency limit, i.e. the highest odd limit to which the edo is consistent, and "Distinct" gives the distinct consistency limit, i.e. the highest odd limit to which the edo is distinctly consistent. The remaining columns give the consistency distance (also called consistency level[note 1]) for every odd limit from 3 to 23.

Edo Consistency limit Consistency distance
Consistent Distinct 3-limit 5-limit 7-limit 9-limit 11-limit 13-limit 15-limit 17-limit 19-limit 21-limit 23-limit
1 3 1 1 0 0 0 0 0 0 0 0 0 0
2 3 1 2 0 0 0 0 0 0 0 0 0 0
3 5 3 2 2 0 0 0 0 0 0 0 0 0
4 7 1 1 1 1 0 0 0 0 0 0 0 0
5 9 3 6 1 1 1 0 0 0 0 0 0 0
6 7 3 1 1 1 0 0 0 0 0 0 0 0
7 5 3 5 1 0 0 0 0 0 0 0 0 0
8 5 3 1 1 0 0 0 0 0 0 0 0 0
9 7 5 1 1 1 0 0 0 0 0 0 0 0
10 7 3 3 1 1 0 0 0 0 0 0 0 0
11 3 3 1 0 0 0 0 0 0 0 0 0 0
12 9 5 25 3 1 1 0 0 0 0 0 0 0
13 3 3 1 0 0 0 0 0 0 0 0 0 0
14 3 3 2 0 0 0 0 0 0 0 0 0 0
15 7 5 2 2 1 0 0 0 0 0 0 0 0
16 7 5 1 1 1 0 0 0 0 0 0 0 0
17 3 3 8 0 0 0 0 0 0 0 0 0 0
18 7 5 1 1 1 0 0 0 0 0 0 0 0
19 9 5 4 4 1 1 0 0 0 0 0 0 0
20 3 3 1 0 0 0 0 0 0 0 0 0 0
21 3 3 1 0 0 0 0 0 0 0 0 0 0
22 11 5 3 2 1 1 1 0 0 0 0 0 0
23 5 5 1 1 0 0 0 0 0 0 0 0 0
24 5 5 12 1 0 0 0 0 0 0 0 0 0
25 5 5 1 1 0 0 0 0 0 0 0 0 0
26 13 5 2 1 1 1 1 1 0 0 0 0 0
27 9 7 2 1 1 1 0 0 0 0 0 0 0
28 5 5 1 1 0 0 0 0 0 0 0 0 0
29 15 5 13 1 1 1 1 1 1 0 0 0 0
30 5 5 1 1 0 0 0 0 0 0 0 0 0
31 11 7 3 3 3 1 1 0 0 0 0 0 0
32 3 3 1 0 0 0 0 0 0 0 0 0 0
33 3 3 1 0 0 0 0 0 0 0 0 0 0
34 5 5 4 4 0 0 0 0 0 0 0 0 0
35 7 7 1 1 1 0 0 0 0 0 0 0 0
36 7 7 8 1 1 0 0 0 0 0 0 0 0
37 7 7 1 1 1 0 0 0 0 0 0 0 0
38 5 5 2 2 0 0 0 0 0 0 0 0 0
39 5 5 2 1 0 0 0 0 0 0 0 0 0
40 3 3 1 0 0 0 0 0 0 0 0 0 0
41 15 9 30 2 2 2 1 1 1 0 0 0 0
42 7 7 1 1 1 0 0 0 0 0 0 0 0
43 7 7 3 1 1 0 0 0 0 0 0 0 0
44 5 5 1 1 0 0 0 0 0 0 0 0 0
45 7 7 1 1 1 0 0 0 0 0 0 0 0
46 13 9 5 2 1 1 1 1 0 0 0 0 0
47 5 5 1 1 0 0 0 0 0 0 0 0 0
48 5 5 6 1 0 0 0 0 0 0 0 0 0
49 7 7 1 1 1 0 0 0 0 0 0 0 0
50 9 7 2 2 1 1 0 0 0 0 0 0 0
51 3 3 2 0 0 0 0 0 0 0 0 0 0
52 3 3 1 0 0 0 0 0 0 0 0 0 0
53 9 9 165 8 1 1 0 0 0 0 0 0 0
54 3 3 1 0 0 0 0 0 0 0 0 0 0
55 5 5 2 1 0 0 0 0 0 0 0 0 0
56 7 7 2 1 1 0 0 0 0 0 0 0 0
57 7 7 1 1 1 0 0 0 0 0 0 0 0
58 17 11 6 1 1 1 1 1 1 1 0 0 0
59 7 7 1 1 1 0 0 0 0 0 0 0 0
60 9 9 5 1 1 1 0 0 0 0 0 0 0
61 5 5 1 1 0 0 0 0 0 0 0 0 0
62 7 7 1 1 1 0 0 0 0 0 0 0 0
63 7 7 3 1 1 0 0 0 0 0 0 0 0
64 3 3 1 0 0 0 0 0 0 0 0 0 0
65 5 5 22 5 0 0 0 0 0 0 0 0 0
66 3 3 1 0 0 0 0 0 0 0 0 0 0
67 3 3 2 0 0 0 0 0 0 0 0 0 0
68 9 9 2 2 2 1 0 0 0 0 0 0 0
69 5 5 1 1 0 0 0 0 0 0 0 0 0
70 9 9 9 1 1 1 0 0 0 0 0 0 0
71 5 5 1 1 0 0 0 0 0 0 0 0 0
72 17 11 4 2 2 2 2 1 1 1 0 0 0
73 7 7 1 1 1 0 0 0 0 0 0 0 0
74 5 5 1 1 0 0 0 0 0 0 0 0 0
75 5 5 3 1 0 0 0 0 0 0 0 0 0
76 7 7 1 1 1 0 0 0 0 0 0 0 0
77 9 9 11 1 1 1 0 0 0 0 0 0 0
78 7 7 1 1 1 0 0 0 0 0 0 0 0
79 5 5 2 1 0 0 0 0 0 0 0 0 0
80 19 11 2 2 1 1 1 1 1 1 1 0 0
81 7 7 1 1 1 0 0 0 0 0 0 0 0
82 9 9 15 1 1 1 0 0 0 0 0 0 0
83 7 7 1 1 1 0 0 0 0 0 0 0 0
84 9 9 3 3 1 1 0 0 0 0 0 0 0
85 3 3 1 0 0 0 0 0 0 0 0 0 0
86 3 3 1 0 0 0 0 0 0 0 0 0 0
87 15 13 4 4 1 1 1 1 1 0 0 0 0
88 7 7 1 1 1 0 0 0 0 0 0 0 0
89 11 11 8 1 1 1 1 0 0 0 0 0 0
90 7 7 1 1 1 0 0 0 0 0 0 0 0
91 9 9 2 1 1 1 0 0 0 0 0 0 0
92 5 5 2 1 0 0 0 0 0 0 0 0 0
93 7 7 1 1 1 0 0 0 0 0 0 0 0
94 23 13 36 1 1 1 1 1 1 1 1 1 1
95 7 7 1 1 1 0 0 0 0 0 0 0 0
96 5 5 3 1 0 0 0 0 0 0 0 0 0
97 5 5 1 1 0 0 0 0 0 0 0 0 0
98 3 3 1 0 0 0 0 0 0 0 0 0 0
99 9 9 5 3 3 2 0 0 0 0 0 0 0

Notes

  1. ↑ This term was coined by Paul Hahn in 1996. See Yahoo! Tuning Group | Consistency generalized.
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