369edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
+RTT table and rank-2 temperaments
Countritonic -> ragitritonic
 
(17 intermediate revisions by 5 users not shown)
Line 1: Line 1:
The '''369 equal divisions of the octave''' divides the octave into 369 [[equal]] parts of 3.252 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
369edo 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 21&109 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.  
369edo shares its [[3/2|perfect fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp.  


369 factors as 3<sup>2</sup> × 41, with subset edos 3, 9, 41, and 123.  
As an equal temperament, it [[tempering out|tempers out]] the [[escapade comma]] and the [[ennealimma]] in the 5-limit; [[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]], so that it supports [[escapade]] in the [[2.3.5.11 subgroup]] and in fact provides the [[optimal patent val]]. It also provides the optimal patent val for the 11-limit {{nowrap| 152 & 217 }} temperament (an escapade extension), the {{nowrap| 130 & 239 }} temperament (a weak escapade extension), 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 [[Tenney–Euclidean tuning|TE-optimal tuning]] of this temperament is [[consistent]] in the 15-integer-limit.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|369}}
{{Harmonics in equal|369|columns=11}}
 
=== Subsets and supersets ===
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 ==
{| 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]]
Line 21: Line 28:
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
| {{Monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
| [{{val| 369 585 857 }}]
| {{Mapping| 369 585 857 }}
| -0.1991
| −0.1991
| 0.1409
| 0.1409
| 4.33
| 4.33
Line 29: Line 36:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
| [{{val| 369 585 857 1036 }}]
| {{Mapping| 369 585 857 1036 }}
| -0.1743
| −0.1743
| 0.1294
| 0.1294
| 3.98
| 3.98
Line 36: Line 43:
| 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
| [{{val| 369 585 857 1036 1277 }}]
| {{Mapping| 369 585 857 1036 1277 }}
| -0.2277
| −0.2277
| 0.1576
| 0.1576
| 4.85
| 4.85
Line 43: Line 50:
| 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
| [{{val| 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
Line 51: Line 58:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
| 1
| 1
| 17\369
| 17\369
| 339.56
| 55.28
| 33/32
| 33/32
| [[Escapade]]
| [[Escapade]]
Line 69: Line 77:
| 864/625
| 864/625
| [[Tritriple]] (5-limit)
| [[Tritriple]] (5-limit)
|-
| 1
| 181\369
| 588.62
| 128/91
| [[Ragitritonic]]
|-
|-
| 9
| 9
Line 74: Line 88:
| 250.41<br>(16.26)
| 250.41<br>(16.26)
| 140/121<br>(100/99)
| 140/121<br>(100/99)
| [[Semiennealimmal]]
| [[Ennealimmapine]]
|-
|-
| 9
| 9
Line 80: Line 94:
| 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
Line 92: Line 106:
| 178.86<br>(3.25)
| 178.86<br>(3.25)
| 567/512<br>(352/351)
| 567/512<br>(352/351)
| [[Hemicounterpyth]]
| [[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


[[Category:Equal divisions of the octave]]
[[Category:Semiporwellismic]]
[[Category:Semiporwellismic]]

Latest revision as of 11:12, 20 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 perfect fifth with 41edo. It has a sharp tendency, with harmonics 3 through 11 all tuned sharp.

As an equal temperament, it tempers out the escapade comma and the ennealimma in the 5-limit; 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, so that it supports escapade in the 2.3.5.11 subgroup and in fact provides the optimal patent val. It also provides the optimal patent val for the 11-limit 152 & 217 temperament (an escapade extension), the 130 & 239 temperament (a weak escapade extension), 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 TE-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)

Subsets and supersets

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 Ragitritonic
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

Retrieved from "https://en.xen.wiki/index.php?title=369edo&oldid=230682"