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315edo: Difference between revisions

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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..."
 
m →Regular temperament properties: quintile isn't defined beyond the 5-limit
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|315}}
{{ED intro}}


== Theory ==
== Theory ==
315et is consistent to the [[7-odd-limit]], tempering 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]].
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, with 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.
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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|-499 315}}
! rowspan="2" | [[Comma list]]
|{{mapping|315 499}}
! rowspan="2" | [[Mapping]]
| 0.3163
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo|-499 315}}
| {{Mapping|315 499}}
| +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, 35595703125/35246833664
| 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>Ratio*
! Associated<br>ratio*
! 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)
| 5\315
|499.05<br>(19.05)
| 19.05
|4/3<br>(81/80)
| 81/80
|[[Pental]]
| [[Quintile]]
|-
|-
|9
| 9
|83\315<br>(13\315)
| 13\315
|316.19<br>(49.52)
| 49.52
|6/5<br>(36/35)
| 36/35
|[[Ennealimmal]]
| [[Ennealimmal]]
|}
|}
 
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
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