315edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|315}} == Theory == 315et is consistent to the 7-odd-limit, tempering out 2401/2400, 4375/4374 and 35595703125/35246833664. Using the 31..." |
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== Theory == | == Theory == | ||
315edo is [[consistent]] to the [[7-odd-limit]] with a flat tendency in the [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]]. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[4375/4374]] and 35595703125/35246833664. Using the 315e [[val]] in the 11-limit ({{val| 315 499 731 884 '''1089''' }}), it tempers out [[385/384]], 1375/1372, 4375/4374 and 644204/643125, [[support]]ing [[beyla]] and [[ennealiminal]]. | |||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
315 factors into 3<sup>2</sup> × 5 × 7, | 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 harmonic 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|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! 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}} | | {{monzo|-499 315}} | ||
|{{mapping|315 499}} | | {{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}} | | {{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 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|2401/2400, 4375/4374, | | 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 | ||
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! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|107\315 | | 107\315 | ||
|407.62 | | 407.62 | ||
|15625/12288 | | 15625/12288 | ||
|[[Ditonic]] | | [[Ditonic]] | ||
|- | |- | ||
|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]] | | [[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 lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||