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

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 (⟨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

315 factors into 3² × 5 × 7, with 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, 35595703125/35246833664	[⟨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
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