369edo: Difference between revisions
+RTT table and rank-2 temperaments |
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{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | == Theory == | ||
369edo | 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 === | === 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 == | == 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 }} | ||
| | | {{Mapping| 369 585 857 }} | ||
| | | −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 }} | ||
| | | {{Mapping| 369 585 857 1036 }} | ||
| | | −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 | ||
| | | {{Mapping| 369 585 857 1036 1277 }} | ||
| | | −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 | ||
| | | {{Mapping| 369 585 857 1036 1277 1366 }} (369f) | ||
| | | −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 | |- | ||
! Generator | ! Periods<br>per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br>ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| 1 | | 1 | ||
| 17\369 | | 17\369 | ||
| | | 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) | ||
| [[ | | [[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) | ||
| [[ | | [[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]] | [[Category:Semiporwellismic]] | ||