31867edo

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31867edo

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Prime factorization	11 × 2897
Step size	0.0376565¢
Fifth	18641\31867 (701.955¢) (convergent)
Semitones (A1:m2)	3019:2396 (113.7¢ : 90.22¢)
Consistency limit	21
Distinct consistency limit	21

31867 equal divisions of the octave (abbreviated 31867edo or 31867ed2), also called 31867-tone equal temperament (31867tet) or 31867 equal temperament (31867et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 31867 equal parts of about 0.0377 ¢ each. Each step represents a frequency ratio of 2^1/31867, or the 31867th root of 2. It is the denominator of the next convergent for log₂3 past 15601, before 79335, and has a fifth which is about 0.00000039 cents compressed.

Theory

31867edo inherits its mapping for the 11-prime from 2897edo, which is the only mapping shared between the two EDO systems- unfortunately, however, this means that the 11-prime is not good for stacking in this system. However, despite this, 31867edo is consistent through the 21-odd-limit, tempering out [305 -106 -59⟩ and [-122 285 -142⟩ in the 5-limit; [-7 30 -9 -7⟩, [51 -13 -1 -10⟩, and [-8 2 -62 53⟩ in the 7-limit; 6576668672/6576582375, 13841287201/13841203200, 11816941917501/11816406250000, and 28247524900000/28245855390489 in the 11-limit; 123201/123200, 1990656/1990625, 72773428/72772425, 1977326743/1977300000, and 6866455078125/6866343676192 in the 13-limit; 194481/194480, 336141/336140, 2000033/2000000, 9765888/9765625, 58464700/58461513, and 114244000/114243723 in the 17-limit; 89376/89375, 104976/104975, 165376/165375, 633556/633555, 709632/709631, and 742586/742577 in the 19-limit.

Prime harmonics

Approximation of prime harmonics in 31867edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0000	-0.0000	+0.0044	+0.0008	+0.0110	+0.0033	-0.0067	+0.0111	-0.0131	-0.0106	-0.0141
	Relative (%)	+0.0	-0.0	+11.7	+2.1	+29.3	+8.8	-17.8	+29.4	-34.9	-28.0	-37.4
Steps (reduced)		31867 (0)	50508 (18641)	73993 (10259)	89462 (25728)	110242 (14641)	117922 (22321)	130255 (2787)	135369 (7901)	144152 (16684)	154809 (27341)	157875 (30407)