315edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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=== 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" | {| 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]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[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 | ||
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| 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 | ||
| Line 46: | Line 47: | ||
=== 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 8ve | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 60: | Line 62: | ||
|- | |- | ||
| 5 | | 5 | ||
| | | 5\315 | ||
| | | 19.05 | ||
| | | 81/80 | ||
| [[ | | [[Quintile]] | ||
|- | |- | ||
| 9 | | 9 | ||
| | | 13\315 | ||
| | | 49.52 | ||
| | | 36/35 | ||
| [[Ennealimmal]] | | [[Ennealimmal]] | ||
|} | |} | ||
<nowiki>* | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||