315edo: Difference between revisions

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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 ===
Line 9: Line 9:


=== 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 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" |[[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
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>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)
| 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 (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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if 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

Latest revision as of 06:25, 21 February 2025

← 314edo 315edo 316edo →
Prime factorization 32 × 5 × 7
Step size 3.80952 ¢ 
Fifth 184\315 (700.952 ¢)
Semitones (A1:m2) 28:25 (106.7 ¢ : 95.24 ¢)
Consistency limit 7
Distinct consistency limit 7

315 equal divisions of the octave (abbreviated 315edo or 315ed2), also called 315-tone equal temperament (315tet) or 315 equal temperament (315et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 315 equal parts of about 3.81 ¢ each. Each step represents a frequency ratio of 21/315, or the 315th root of 2.

Theory

315edo is consistent to the 7-odd-limit with a flat tendency in the harmonics 3, 5, and 7. The equal temperament tempers out 2401/2400, 4375/4374 and 35595703125/35246833664. Using the 315e val in the 11-limit (⟨315 ​499 ​731​ 884​ 1089]), it tempers out 385/384, 1375/1372, 4375/4374 and 644204/643125, supporting beyla and ennealiminal.

Odd harmonics

Approximation of odd harmonics in 315edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.00 -1.55 -1.21 +1.80 +1.06 +1.38 +1.26 +1.71 -0.37 +1.60 +0.30
Relative (%) -26.3 -40.7 -31.7 +47.4 +27.9 +36.1 +32.9 +44.9 -9.7 +42.0 +7.8
Steps
(reduced) 		499
(184) 		731
(101) 		884
(254) 		999
(54) 		1090
(145) 		1166
(221) 		1231
(286) 		1288
(28) 		1338
(78) 		1384
(124) 		1425
(165)

Subsets and supersets

Since 315 factors into 32 × 5 × 7, 315edo has subset 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-499 315⟩ [⟨315 499]] +0.3163 0.3164 8.31
2.3.5 [-27 -2 13⟩, [-28 25 -5⟩ [⟨315 499 731]] +0.4337 0.3071 8.06
2.3.5.7 2401/2400, 4375/4374, [-21 6 11 -5⟩ [⟨315 499 731 884]] +0.4328 0.2659 6.98

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 107\315 407.62 15625/12288 Ditonic
5 131\315
(5\315) 		499.05
(19.05) 		4/3
(81/80) 		Pental (5-limit)
9 83\315
(13\315) 		316.19
(49.52) 		6/5
(36/35) 		Ennealimmal

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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