315edo

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← 314edo315edo316edo →
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 3.810 ¢ 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 -41 -32 +47 +28 +36 +33 +45 -10 +42 +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 it is distinct