369edo: Difference between revisions
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== Theory == | == Theory == | ||
369 = 9 × 41, and it shares the fifth with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. It tempers out [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s 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. | 369 = 9 × 41, and it shares the fifth with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. It tempers out [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s 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. | 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. | ||
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{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
| Line 73: | Line 73: | ||
| 864/625 | | 864/625 | ||
| [[Tritriple]] (5-limit) | | [[Tritriple]] (5-limit) | ||
|- | |||
| 1 | |||
| 181\369 | |||
| 588.62 | |||
| 128/91 | |||
| [[Countritonic]] | |||
|- | |- | ||
| 9 | | 9 | ||
Revision as of 09:40, 6 September 2023
| ← 368edo | 369edo | 370edo → |
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.
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) | |
Divisors
Since 369 factors into 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 (Reduced) |
Cents (Reduced) |
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 | Countritonic |
| 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) |
Hemicountercomp |