# 315edo

 ← 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 it is distinct