8404edo
| ← 8403edo | 8404edo | 8405edo → |
8404 equal divisions of the octave (abbreviated 8404edo or 8404ed2), also called 8404-tone equal temperament (8404tet) or 8404 equal temperament (8404et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 8404 equal parts of about 0.143 ¢ each. Each step represents a frequency ratio of 21/8404, or the 8404th root of 2.
8404edo is consistent to the 13-odd-limit, but both harmonics 5 and 13 are about halfway between its steps. It is perhaps more interesting as every other step of the monstrous 16808edo, with the same extraordinary accuracy in the 2.3.25.7.11.17.23.31 subgroup. Moreover, it can be used adaptively to mimic 16808edo by alternating the sharp and flat mappings of the inaccurate primes, if you believe the full 16808edo is not worth going the extra mile. 8269edo and 8539edo are similarly sized edos that legit approximate JI without this trick, though they do best in the 23-limit, not the 31-limit.
Like 16808edo, 8404edo's perfect fifth comes from 2101edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0035 | -0.0691 | -0.0015 | -0.0090 | -0.0707 | -0.0054 | +0.0596 | -0.0021 | -0.0674 | -0.0094 |
| Relative (%) | +0.0 | -2.5 | -48.4 | -1.1 | -6.3 | -49.5 | -3.8 | +41.7 | -1.5 | -47.2 | -6.6 | |
| Steps (reduced) |
8404 (0) |
13320 (4916) |
19513 (2705) |
23593 (6785) |
29073 (3861) |
31098 (5886) |
34351 (735) |
35700 (2084) |
38016 (4400) |
40826 (7210) |
41635 (8019) | |
Subsets and supersets
Since 8404 factors into primes as 22 × 11 × 191, 8404edo contains subset edos 2, 4, 11, 22, 44, 191, 382, 764, 2101, and 4202. One step of 8404edo is exactly 2 jinns.