Minimal consistent EDOs

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%. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135.

Odd limit	Smallest consistent edo*	Smallest distinctly consistent edo	Smallest purely consistent** edo
1	1	1	1
3		3	2
5	3	9	3
7	4	27	10
9	5	41	41
11	22	58
13	26	87	46
15	29	111	87
17	58	149	311
19	80	217
21	94	282
23
25	282	388
27
29		1323
31	311	1600
33
35
37
39		2554
41
43	17461	17461	20567
45
47	20567	20567
49			459944
51
53			1705229
55
57
59	253389	253389	3159811
61	625534	625534
63
65
67			7317929
69	759630	759630	8595351
71
73			27783092
75	2157429	2157429	34531581
77
79	2901533	2901533	50203972
81
83
85
87
89
91
93
95
97			1297643131
99
101			3888109922
103
105
107			13805152233
109			27218556026
111
113
115
117
119			42586208631
121
123
125
127
129
131			93678217813
133	70910024	70910024
135

* apart from 0edo

** purely consistent is an ^{[idiosyncratic term]}

*** purely consistent to the 137-odd-limit

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

• OEIS: Equal divisions of the octave with progressively increasing consistency levels (OEIS)
• OEIS: Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit (OEIS)
• OEIS: Equal divisions of the octave with nondecreasing consistency levels. (OEIS)
• OEIS: Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit (OEIS)