31867edo
| ← 31866edo | 31867edo | 31868edo → |
(convergent)
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 0.038 ¢ 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
| 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) | |