31867edo

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← 31866edo31867edo31868edo →
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 21/31867, 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 +12 +2 +29 +9 -18 +29 -35 -28 -37
Steps
(reduced)
31867
(0)
50508
(18641)
73993
(10259)
89462
(25728)
110242
(14641)
117922
(22321)
130255
(2787)
135369
(7901)
144152
(16684)
154809
(27341)
157875
(30407)