Consistency limits of small EDOs: Difference between revisions

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AAn [[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 [[Consistent#Consistency to distance d|"Consistency distance"]] (also called "consistency level"<ref>This term was coined by [[Paul Hahn]] in 1996: https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html</ref>) 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 220: Line 220:
|}
|}


[[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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