369edo
The 369 equal divisions of the octave (369edo), or the 369(-tone) equal temperament (369tet, 369et) when viewed from a regular temperament perspective, divides the octave into 369 equal parts of about 3.25 cents each.
Theory
369 = 9 × 41, and it shares the fifth with 41edo. It has a sharp tendency, with harmonics 3 through 11 all tuned sharp. It tempers out 2401/2400 and 4375/4374 in the 7-limit, so that it supports the ennealimmal temperament; in the 11-limit, 4000/3993, 5632/5625 and 16384/16335. It provides the optimal patent val for the 11-limit 130&239 temperament, the 65&152 temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as the no-7 subgroup version of it.
Extension to the 13-limit is viable by the 369f val, tempering out 1575/1573, 2080/2079, 2200/2197, and 3584/3575. The optimal tuning of this temperament is consistent in the 15-integer-limit.
369 has subset edos 3, 9, 41, and 123.
Prime harmonics
Script error: No such module "primes_in_edo".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [32 -7 -9⟩, [1 -27 18⟩ | [⟨369 585 857]] | -0.1991 | 0.1409 | 4.33 |
| 2.3.5.7 | 2401/2400, 4375/4374, [32 -7 -9⟩ | [⟨369 585 857 1036]] | -0.1743 | 0.1294 | 3.98 |
| 2.3.5.7.11 | 2401/2400, 4000/3993, 4375/4374, 5632/5625 | [⟨369 585 857 1036 1277]] | -0.2277 | 0.1576 | 4.85 |
| 2.3.5.7.11.13 | 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575 | [⟨369 585 857 1036 1277 1366]] (369f) | -0.2685 | 0.1703 | 5.24 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 17\369 | 55.28 | 33/32 | Escapade |
| 1 | 172\369 | 559.35 | 864/625 | Tritriple (5-limit) |
| 9 | 77\369 (5\369) |
250.41 (16.26) |
140/121 (100/99) |
Semiennealimmal |
| 9 | 97\369 (15\369) |
315.45 (48.78) |
6/5 (36/35) |
Ennealimmal |
| 9 | 68\369 (14\369) |
221.14 (45.53) |
25/22 (77/75) |
Quadraennealimmal |
| 41 | 55\369 (1\369) |
178.86 (3.25) |
567/512 (352/351) |
Hemicounterpyth |