369edo: Difference between revisions
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+link to countritonic |
Cleanup; clarify the title row of the rank-2 temp table; -redundant categories |
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== Theory == | == Theory == | ||
369 = 9 × 41, and | 369 = 9 × 41, and 369edo shares its [[3/2|fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out|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 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 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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=== Divisors === | === Divisors === | ||
Since 369 factors into | Since 369 factors into {{factorization|369}}, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| 2.3.5 | | 2.3.5 | ||
| {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }} | | {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }} | ||
| | | {{mapping| 369 585 857 }} | ||
| -0.1991 | | -0.1991 | ||
| 0.1409 | | 0.1409 | ||
| Line 33: | Line 33: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }} | | 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }} | ||
| | | {{mapping| 369 585 857 1036 }} | ||
| -0.1743 | | -0.1743 | ||
| 0.1294 | | 0.1294 | ||
| Line 40: | Line 40: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 4000/3993, 4375/4374, 5632/5625 | | 2401/2400, 4000/3993, 4375/4374, 5632/5625 | ||
| | | {{mapping| 369 585 857 1036 1277 }} | ||
| -0.2277 | | -0.2277 | ||
| 0.1576 | | 0.1576 | ||
| Line 47: | Line 47: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575 | | 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575 | ||
| | | {{mapping| 369 585 857 1036 1277 1366 }} (369f) | ||
| -0.2685 | | -0.2685 | ||
| 0.1703 | | 0.1703 | ||
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|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 104: | Line 104: | ||
| [[Hemicountercomp]] | | [[Hemicountercomp]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Semiporwellismic]] | [[Category:Semiporwellismic]] | ||
Revision as of 14:54, 9 November 2023
| ← 368edo | 369edo | 370edo → |
Theory
369 = 9 × 41, and 369edo shares its fifth with 41edo. It has a sharp tendency, with harmonics 3 through 11 all tuned sharp. The equal temperament 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 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 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* | 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 | 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 |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct