369edo: Difference between revisions

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== Theory ==
== Theory ==
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.  
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 {{nowrap|130 & 239}} temperament, {{nowrap|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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| {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
| {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
| {{mapping| 369 585 857 }}
| {{mapping| 369 585 857 }}
| −0.1991
| −0.1991
| 0.1409
| 0.1409
| 4.33
| 4.33
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| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
| {{mapping| 369 585 857 1036 }}
| {{mapping| 369 585 857 1036 }}
| −0.1743
| −0.1743
| 0.1294
| 0.1294
| 3.98
| 3.98
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| 2401/2400, 4000/3993, 4375/4374, 5632/5625
| 2401/2400, 4000/3993, 4375/4374, 5632/5625
| {{mapping| 369 585 857 1036 1277 }}
| {{mapping| 369 585 857 1036 1277 }}
| −0.2277
| −0.2277
| 0.1576
| 0.1576
| 4.85
| 4.85
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| 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)
| {{mapping| 369 585 857 1036 1277 1366 }} (369f)
| −0.2685
| −0.2685
| 0.1703
| 0.1703
| 5.24
| 5.24
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| [[Hemicountercomp]]
| [[Hemicountercomp]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Semiporwellismic]]
[[Category:Semiporwellismic]]

Revision as of 19:24, 15 January 2025

← 368edo 369edo 370edo →
Prime factorization 32 × 41
Step size 3.25203 ¢ 
Fifth 216\369 (702.439 ¢) (→ 24\41)
Semitones (A1:m2) 36:27 (117.1 ¢ : 87.8 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

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, 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

Approximation of prime harmonics in 369edo
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

Table of rank-2 temperaments by generator
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 distinct

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