369edo
| ← 368edo | 369edo | 370edo → |
369 equal divisions of the octave (abbreviated 369edo or 369ed2), also called 369-tone equal temperament (369tet) or 369 equal temperament (369et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 369 equal parts of about 3.25 ¢ each. Each step represents a frequency ratio of 21/369, or the 369th root of 2.
Theory
369edo shares its perfect fifth with 41edo. It has a sharp tendency, with harmonics 3 through 11 all tuned sharp.
As an equal temperament, it tempers out the escapade comma and the ennealimma in the 5-limit; 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, so that it supports escapade in the 2.3.5.11 subgroup and in fact provides the optimal patent val. It also provides the optimal patent val for the 11-limit 152 & 217 temperament (an escapade extension), the 130 & 239 temperament (a weak escapade extension), and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.
Extension to the 13-limit is viable by the 369f val, tempering out 1575/1573, 2080/2079, 2200/2197, and 3584/3575. The TE-optimal tuning of this temperament is consistent in the 15-integer-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.48 | +0.68 | +0.28 | +1.53 | -1.50 | -0.89 | -1.58 | -0.63 | +1.32 | -0.32 |
| Relative (%) | +0.0 | +14.9 | +20.9 | +8.6 | +47.0 | -46.2 | -27.4 | -48.5 | -19.4 | +40.5 | -9.8 | |
| Steps (reduced) |
369 (0) |
585 (216) |
857 (119) |
1036 (298) |
1277 (170) |
1365 (258) |
1508 (32) |
1567 (91) |
1669 (193) |
1793 (317) |
1828 (352) | |
Subsets and supersets
Since 369 factors into primes as 32 × 41, 369edo has subset edos 3, 9, 41, and 123.
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 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 17\369 | 55.28 | 33/32 | Escapade |
| 1 | 172\369 | 559.35 | 864/625 | Tritriple (5-limit) |
| 1 | 181\369 | 588.62 | 128/91 | Ragitritonic |
| 9 | 77\369 (5\369) |
250.41 (16.26) |
140/121 (100/99) |
Ennealimmapine |
| 9 | 97\369 (15\369) |
315.45 (48.78) |
6/5 (36/35) |
Ennealimmal / enneabiotic |
| 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) |
Hemicountercomp |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct