315edo: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
m →Regular temperament properties: quintile isn't defined beyond the 5-limit |
||
| (6 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 315 factors into 3<sup>2</sup> × 5 × 7, 315edo has subset edos {{EDOs| 3, 5, 7, 9, 15, 21, 35, 45, 63, and 105 }}. [[945edo]], which triples it, gives a good correction to the | Since 315 factors into 3<sup>2</sup> × 5 × 7, 315edo has subset edos {{EDOs| 3, 5, 7, 9, 15, 21, 35, 45, 63, and 105 }}. [[945edo]], which triples it, gives a good correction to the harmonics 3, 5, 7, and 11. | ||
== Regular temperament properties == | == 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 | | 2.3 | ||
| {{ | | {{Monzo|-499 315}} | ||
| {{ | | {{Mapping|315 499}} | ||
| 0.3163 | | +0.3163 | ||
| 0.3164 | | 0.3164 | ||
| 8.31 | | 8.31 | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| -27 -2 13 }}, {{monzo| -28 25 -5 }} | ||
| {{ | | {{Mapping| 315 499 731 }} | ||
| 0.4337 | | +0.4337 | ||
| 0.3071 | | 0.3071 | ||
| 8.06 | | 8.06 | ||
| Line 30: | Line 39: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 4375/4374, {{monzo| -21 6 11 -5 }} | | 2401/2400, 4375/4374, {{monzo| -21 6 11 -5 }} | ||
| {{ | | {{Mapping| 315 499 731 884 }} | ||
| 0.4328 | | +0.4328 | ||
| 0.2659 | | 0.2659 | ||
| 6.98 | | 6.98 | ||
|} | |||
=== Rank-2 temperaments === | === 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 | | 1 | ||
| Line 46: | Line 62: | ||
|- | |- | ||
| 5 | | 5 | ||
| | | 5\315 | ||
| | | 19.05 | ||
| | | 81/80 | ||
| [[ | | [[Quintile]] | ||
|- | |- | ||
| 9 | | 9 | ||
| | | 13\315 | ||
| | | 49.52 | ||
| | | 36/35 | ||
| [[Ennealimmal]] | | [[Ennealimmal]] | ||
|} | |||
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | |||