315edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[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 | ||
| Line 24: | Line 33: | ||
| {{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 | ||
| Line 31: | Line 40: | ||
| 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 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 56: | Line 72: | ||
| 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 | |||
Latest revision as of 06:25, 21 February 2025
| ← 314edo | 315edo | 316edo → |
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
| 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
| 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