315edo: Difference between revisions
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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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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]] (%) | ||
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| {{monzo|-499 315}} | | {{monzo|-499 315}} | ||
| {{mapping|315 499}} | | {{mapping|315 499}} | ||
| 0.3163 | | +0.3163 | ||
| 0.3164 | | 0.3164 | ||
| 8.31 | | 8.31 | ||
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| {{monzo| -27 -2 13 }}, {{monzo| -28 25 -5 }} | | {{monzo| -27 -2 13 }}, {{monzo| -28 25 -5 }} | ||
| {{mapping| 315 499 731 }} | | {{mapping| 315 499 731 }} | ||
| 0.4337 | | +0.4337 | ||
| 0.3071 | | 0.3071 | ||
| 8.06 | | 8.06 | ||
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| 2401/2400, 4375/4374, {{monzo| -21 6 11 -5 }} | | 2401/2400, 4375/4374, {{monzo| -21 6 11 -5 }} | ||
| {{mapping| 315 499 731 884 }} | | {{mapping| 315 499 731 884 }} | ||
| 0.4328 | | +0.4328 | ||
| 0.2659 | | 0.2659 | ||
| 6.98 | | 6.98 | ||
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=== 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 | ||
|- | |- | ||
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|- | |- | ||
| 5 | | 5 | ||
| 131\315<br>(5\315) | | 131\315<br />(5\315) | ||
| 499.05<br>(19.05) | | 499.05<br />(19.05) | ||
| 4/3<br>(81/80) | | 4/3<br />(81/80) | ||
| [[Pental (temperament)|Pental]] (5-limit) | | [[Pental (temperament)|Pental]] (5-limit) | ||
|- | |- | ||
| 9 | | 9 | ||
| 83\315<br>(13\315) | | 83\315<br />(13\315) | ||
| 316.19<br>(49.52) | | 316.19<br />(49.52) | ||
| 6/5<br>(36/35) | | 6/5<br />(36/35) | ||
| [[Ennealimmal]] | | [[Ennealimmal]] | ||
|} | |} | ||
<nowiki>* | <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 | ||