Minimal consistent EDOs

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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 ⁠^{[idiosyncratic term ]} if its relative errors on odd harmonics up to and including q never exceed 25%. It is accurately consistent⁠ ⁠^{[idiosyncratic term ]} if the edo is consistent to distance 2, or alternatively put, every q-odd-limit interval in the edo has at most 25% relative error. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135. Odd limits of 2ⁿ − 1 are highlighted.

Smallest consistent EDOs per odd limit

Odd limit	Smallest consistent edo*	Smallest distinctly consistent edo	Smallest purely consistent edo	Smallest accurately consistent edo	Smallest distinctly accurate edo
1	1	1	1	1	1
3	1	3	2	2	3
5	3	9	3	3	12
7	4	27	10	31	31
9	5	41	41	41	41
11	22	58	41	72	72
13	26	87	46	270	270
15	29	111	87	494	494
17	58	149	311	3395	3395
19	80	217	311	8539	8539
21	94	282	311	8539	8539
23	94	282	311	16808	16808
25	282	388	311	16808	16808
27	282	388	311	16808	16808
29	282	1323	311	16808	16808
31	311	1600	311	16808	16808
33	311	1600	311	16808	16808
35	311	1600	311	16808	16808
37	311	1600	311	324296	324296
39	311	2554	311	2398629	2398629
41	311	2554	311	19164767	19164767
43	17461	17461	20567	19735901	19735901
45	17461	17461	20567	19735901	19735901
47	20567	20567	20567	152797015	152797015
49	20567	20567	459944
51	20567	20567	459944
53	20567	20567	1705229
55	20567	20567	1705229
57	20567	20567	1705229
59	253389	253389	3159811
61	625534	625534	3159811
63	625534	625534	3159811
65	625534	625534	3159811
67	625534	625534	7317929
69	759630	759630	8595351
71	759630	759630	8595351
73	759630	759630	27783092
75	2157429	2157429	34531581
77	2157429	2157429	34531581
79	2901533	2901533	50203972
81	2901533	2901533	50203972
83	2901533	2901533	50203972
85	2901533	2901533	50203972
87	2901533	2901533	50203972
89	2901533	2901533	50203972
91	2901533	2901533	50203972
93	2901533	2901533	50203972
95	2901533	2901533	50203972
97	2901533	2901533	1297643131
99	2901533	2901533	1297643131
101	2901533	2901533	3888109922
103	2901533	2901533	3888109922
105	2901533	2901533	3888109922
107	2901533	2901533	13805152233
109	2901533	2901533	27218556026
111	2901533	2901533	27218556026
113	2901533	2901533	27218556026
115	2901533	2901533	27218556026
117	2901533	2901533	27218556026
119	2901533	2901533	42586208631
121	2901533	2901533	42586208631
123	2901533	2901533	42586208631
125	2901533	2901533	42586208631
127	2901533	2901533	42586208631
129	2901533	2901533	42586208631
131	2901533	2901533	93678217813**
133	70910024	70910024	93678217813
135	70910024	70910024	93678217813

* Apart from 0edo

** 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)

See also

• Consistency limits of small EDOs