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

*apart from 0edo

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)