369edo: Difference between revisions
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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. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|369|columns=11}} | |||
=== | === Divisors === | ||
{{ | Since 369 factors into 3<sup>2</sup> × 41, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| 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]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 55: | Line 56: | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 95: | Line 96: | ||
| 178.86<br>(3.25) | | 178.86<br>(3.25) | ||
| 567/512<br>(352/351) | | 567/512<br>(352/351) | ||
| [[ | | [[Hemicountercomp]] | ||
|} | |} | ||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Semiporwellismic]] | [[Category:Semiporwellismic]] | ||
Revision as of 13:45, 22 December 2022
| ← 368edo | 369edo | 370edo → |
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.
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) |
| 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 |