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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
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=== Divisors ===
=== Divisors ===
Since 369 factors into {{factorization|369}}, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}.  
Since 369 factors into primes as {{nowrap| 3<sup>2</sup> × 41 }}, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}.  


== Regular temperament properties ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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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 }}
| {{Mapping| 369 585 857 }}
| −0.1991
| −0.1991
| 0.1409
| 0.1409
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| 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 }}
| {{Mapping| 369 585 857 1036 }}
| −0.1743
| −0.1743
| 0.1294
| 0.1294
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| 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 }}
| {{Mapping| 369 585 857 1036 1277 }}
| −0.2277
| −0.2277
| 0.1576
| 0.1576
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| 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)
| {{Mapping| 369 585 857 1036 1277 1366 }} (369f)
| −0.2685
| −0.2685
| 0.1703
| 0.1703
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | 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
|-
|-
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|-
|-
| 9
| 9
| 77\369<br />(5\369)
| 77\369<br>(5\369)
| 250.41<br />(16.26)
| 250.41<br>(16.26)
| 140/121<br />(100/99)
| 140/121<br>(100/99)
| [[Semiennealimmal]]
| [[Ennealimmapine]]
|-
|-
| 9
| 9
| 97\369<br />(15\369)
| 97\369<br>(15\369)
| 315.45<br />(48.78)
| 315.45<br>(48.78)
| 6/5<br />(36/35)
| 6/5<br>(36/35)
| [[Ennealimmal]]
| [[Ennealimmal]] / enneabiotic
|-
|-
| 9
| 9
| 68\369<br />(14\369)
| 68\369<br>(14\369)
| 221.14<br />(45.53)
| 221.14<br>(45.53)
| 25/22<br />(77/75)
| 25/22<br>(77/75)
| [[Quadraennealimmal]]
| [[Quadraennealimmal]]
|-
|-
| 41
| 41
| 55\369<br />(1\369)
| 55\369<br>(1\369)
| 178.86<br />(3.25)
| 178.86<br>(3.25)
| 567/512<br />(352/351)
| 567/512<br>(352/351)
| [[Hemicountercomp]]
| [[Hemicountercomp]]
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


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

Revision as of 08:53, 19 May 2026

← 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

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 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 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

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) 		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

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