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 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.


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Revision as of 21:36, 28 July 2022

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.

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
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