Consistency limits of small EDOs: Difference between revisions

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An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[direct approximation]] of every element of s is the closest N-EDO approximation. It is [[distinctly consistent]] if every element of s is mapped to a distinct value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly consistent, respectively. 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", see [[Consistent#Consistency to distance ''d'']] (also called Consistency level) for every odd limit from 3 to 23.
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 sortable mw-collapsible" style="text-align:right"
{| class="wikitable sortable mw-collapsible right-all"
|-
|-
! rowspan="2" | EDO
! rowspan="2" | Edo
! colspan="2" | Consistency limit
! colspan="2" | Consistency limit
! colspan="11" | Consistency distance
! colspan="11" | Consistency distance
Line 9: Line 9:
! Consistent
! Consistent
! Distinct
! Distinct
! 3-limit
! 3-limit
! 5-limit
! 5-limit
! 7-limit
! 7-limit
! 9-limit
! 9-limit
! 11-limit
! 11-limit
! 13-limit
! 13-limit
Line 21: Line 21:
! 23-limit
! 23-limit
|-
|-
| 1 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 1 || style="background-color: #fefefe;"| 3 || 1 || style="background-color: #fefefe;"| 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
| 2 || style="background-color: #fefefe;"| 3 || 1 || style="background-color: #fefefe;"| 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
| 3 || style="background-color: #fefefe;"| 5 || 3 || 2 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 4 || 7 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 4 || style="background-color: #fefefe;"| 7 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 5 || 9 || 3 || 6 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 5 || style="background-color: #fefefe;"| 9 || 3 || 6 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 6 || 7 || 3 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 6 || style="background-color: #fefefe;"| 7 || 3 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 7 || 5 || 3 || 5 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 7 || style="background-color: #fefefe;"| 5 || 3 || 5 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 8 || 5 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 8 || style="background-color: #fefefe;"| 5 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 9 || 7 || 5 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 9 || style="background-color: #fefefe;"| 7 || 5 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 10 || 7 || 3 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 10 || style="background-color: #fefefe;"| 7 || 3 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 11 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 11 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 12 || style="background-color: #fefefe;"| 9 || 5 || 25 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 13 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 13 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 14 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 15 || style="background-color: #fefefe;"| 7 || 5 || 2 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 16 || 7 || 5 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 16 || style="background-color: #fefefe;"| 7 || 5 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 17 || 3 || 3 || 8 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 17 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 18 || style="background-color: #fefefe;"| 7 || 5 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 19 || 9 || 5 || 4 || 4 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 19 || style="background-color: #fefefe;"| 9 || 5 || 4 || 4 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 20 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 20 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 21 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 22 || style="background-color: #fefefe;"| 11 || 5 || 3 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 23 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 23 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 24 || 5 || 5 || 12 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 24 || style="background-color: #fefefe;"| 5 || 5 || 12 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 25 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 25 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 26 || 13 || 5 || 2 || 1 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0
| 26 || style="background-color: #fefefe;"| 13 || 5 || 2 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0
|-
|-
| 27 || 9 || 7 || 2 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 27 || style="background-color: #fefefe;"| 9 || 7 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 28 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 28 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 29 || 15 || 5 || 13 || 1 || 1 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
| 29 || style="background-color: #fefefe;"| 15 || 5 || 13 || 1 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0
|-
|-
| 30 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 30 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 31 || 11 || 7 || 3 || 3 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0
| 31 || style="background-color: #fefefe;"| 11 || 7 || 3 || 3 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 32 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 32 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 33 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 34 || style="background-color: #fefefe;"| 5 || 5 || 4 || style="background-color: #fefefe;"| 4 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 35 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 35 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 36 || 7 || 7 || 8 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 36 || style="background-color: #fefefe;"| 7 || 7 || 8 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 37 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 37 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 38 || 5 || 5 || 2 || 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 38 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 39 || 5 || 5 || 2 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 39 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 40 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 40 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 41 || style="background-color: #fefefe;"| 15 || 9 || 30 || 2 || 2 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0
|-
|-
| 42 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 42 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 43 || 7 || 7 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 43 || style="background-color: #fefefe;"| 7 || 7 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 44 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 44 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 45 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 45 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 46 || 13 || 9 || 5 || 2 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0
| 46 || style="background-color: #fefefe;"| 13 || 9 || 5 || 2 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0
|-
|-
| 47 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 47 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 48 || 5 || 5 || 6 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 48 || style="background-color: #fefefe;"| 5 || 5 || 6 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 49 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 49 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 50 || 9 || 7 || 2 || 2 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 50 || style="background-color: #fefefe;"| 9 || 7 || 2 || 2 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 51 || 3 || 3 || 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 51 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 52 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 53 || style="background-color: #fefefe;"| 9 || 9 || 165 || 8 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 54 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 54 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 55 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 56 || 7 || 7 || 2 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 56 || style="background-color: #fefefe;"| 7 || 7 || 2 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 57 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 57 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 58 || 17 || 11 || 6 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 0 || 0 || 0
| 58 || style="background-color: #fefefe;"| 17 || 11 || 6 || 1 || 1 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0
|-
|-
| 59 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 59 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 60 || 9 || 9 || 5 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 60 || style="background-color: #fefefe;"| 9 || 9 || 5 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 61 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 61 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 62 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 62 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 63 || 7 || 7 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 63 || style="background-color: #fefefe;"| 7 || 7 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 64 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 64 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 65 || style="background-color: #fefefe;"| 5 || 5 || 22 || style="background-color: #fefefe;"| 5 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 66 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 66 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 67 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 68 || style="background-color: #fefefe;"| 9 || 9 || 2 || 2 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 69 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 69 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 70 || 9 || 9 || 9 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 70 || style="background-color: #fefefe;"| 9 || 9 || 9 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 71 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 71 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 72 || 17 || 11 || 4 || 2 || 2 || 2 || 2 || 1 || 1 || 1 || 0 || 0 || 0
| 72 || style="background-color: #fefefe;"| 17 || 11 || 4 || 2 || 2 || 2 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0
|-
|-
| 73 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 73 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 74 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 74 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 75 || 5 || 5 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 75 || style="background-color: #fefefe;"| 5 || 5 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 76 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 76 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 77 || 9 || 9 || 11 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 77 || style="background-color: #fefefe;"| 9 || 9 || 11 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 78 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 78 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 79 || 5 || 5 || 2 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 79 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 80 || 19 || 11 || 2 || 2 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 0 || 0
| 80 || style="background-color: #fefefe;"| 19 || 11 || 2 || 2 || 1 || 1 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0
|-
|-
| 81 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 81 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 82 || 9 || 9 || 15 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 82 || style="background-color: #fefefe;"| 9 || 9 || 15 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 83 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 83 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 84 || 9 || 9 || 3 || 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 84 || style="background-color: #fefefe;"| 9 || 9 || 3 || 3 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 85 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 85 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 86 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 87 || style="background-color: #fefefe;"| 15 || 13 || 4 || 4 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0
|-
|-
| 88 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 88 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 89 || 11 || 11 || 8 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0
| 89 || style="background-color: #fefefe;"| 11 || 11 || 8 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 90 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 90 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 91 || 9 || 9 || 2 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 91 || style="background-color: #fefefe;"| 9 || 9 || 2 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 92 || 5 || 5 || 2 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 92 || style="background-color: #fefefe;"| 5 || 5 || 2 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 93 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 93 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 94 || 23 || 13 || 36 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 94 || style="background-color: #fefefe;"| 23 || 13 || 36 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || style="background-color: #fefefe;"| 1
|-
|-
| 95 || 7 || 7 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 95 || style="background-color: #fefefe;"| 7 || 7 || 1 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 96 || 5 || 5 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 96 || style="background-color: #fefefe;"| 5 || 5 || 3 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 97 || 5 || 5 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 97 || style="background-color: #fefefe;"| 5 || 5 || 1 || style="background-color: #fefefe;"| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| 98 || 3 || 3 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 98 || style="background-color: #fefefe;"| 3 || 3 || style="background-color: #fefefe;"| 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
| 99 || style="background-color: #fefefe;"| 9 || 9 || 5 || 3 || 3 || style="background-color: #fefefe;"| 2 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|}
|}


[[Category:Theory]]
== Notes ==
<references group="note"/>
 
[[Category:Mapping]]
[[Category:Mapping]]
[[Category:Consistency]]
[[Category:Consistency]]
[[Category:Odd limit]]
[[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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