Xenharmonic Wiki

369edo: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>FREEZE
No edit summary
Countritonic -> ragitritonic
 
(19 intermediate revisions by 5 users not shown)
Line 1: Line 1:
The 369 equal division divides the octave into 369 equal parts of 3.252 cents each. It tempers out 2401/2400 and 4375/4374 in the 7-limit, so that it supports ennealimmal temperament; in the 11-limit, 4000/3993 and 5632/5625. 369 provides the [[Optimal_patent_val|optimal patent val]] for both the 11-limit 21&109 temperament and the 65&152 temperament. 369 factors as 3^2 * 41.
{{Infobox ET}}
{{ED intro}}
 
== Theory ==
369edo shares its [[3/2|perfect fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp.  
 
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 ===
{{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 ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| {{Monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
| {{Mapping| 369 585 857 }}
| −0.1991
| 0.1409
| 4.33
|-
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
| {{Mapping| 369 585 857 1036 }}
| −0.1743
| 0.1294
| 3.98
|-
| 2.3.5.7.11
| 2401/2400, 4000/3993, 4375/4374, 5632/5625
| {{Mapping| 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
| {{Mapping| 369 585 857 1036 1277 1366 }} (369f)
| −0.2685
| 0.1703
| 5.24
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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<br>(5\369)
| 250.41<br>(16.26)
| 140/121<br>(100/99)
| [[Ennealimmapine]]
|-
| 9
| 97\369<br>(15\369)
| 315.45<br>(48.78)
| 6/5<br>(36/35)
| [[Ennealimmal]] / enneabiotic
|-
| 9
| 68\369<br>(14\369)
| 221.14<br>(45.53)
| 25/22<br>(77/75)
| [[Quadraennealimmal]]
|-
| 41
| 55\369<br>(1\369)
| 178.86<br>(3.25)
| 567/512<br>(352/351)
| [[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:Semiporwellismic]]
Retrieved from "https://en.xen.wiki/w/369edo"