316edo

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316edo

317edo →

Prime factorization	22 × 79
Step size	3.79747 ¢
Fifth	185\316 (702.532 ¢)
Semitones (A1:m2)	31:23 (117.7 ¢ : 87.34 ¢)
Consistency limit	11
Distinct consistency limit	11

Theory

While not highly accurate for its size, 316et is the point where a few important temperaments meet, and is distinctly consistent in the 11-odd-limit. It tempers out the parakleisma, [8 14 -13⟩, the undim comma, [41 -20 -4⟩, and the maquila comma, [49 -6 -17⟩ in the 5-limit; 3136/3125, 4375/4374, 10976/10935 in the 7-limit; 3025/3024, 3388/3375, 9801/9800, and 14641/14580 in the 11-limit; and using the patent val, 1716/1715, 2080/2079, 2197/2187, 4096/4095, 4225/4224, 6656/6655, and 10648/10647 in the 13-limit; notably supporting Abigail and semiparakleismic.

It provides the optimal patent val for the rank-4 temperament tempering out 3388/3375, and triglav, which also tempers out 3025/3024.

Prime harmonics

Approximation of prime harmonics in 316edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.58	+1.03	-0.47	-0.69	-1.29	+1.37	-1.31	-1.69	-0.46	+1.80
	Relative (%)	+0.0	+15.2	+27.1	-12.4	-18.0	-33.9	+36.2	-34.5	-44.6	-12.2	+47.4
Steps (reduced)		316 (0)	501 (185)	734 (102)	887 (255)	1093 (145)	1169 (221)	1292 (28)	1342 (78)	1429 (165)	1535 (271)	1566 (302)

Subsets and supersets

316 factors into 2² × 79, with subset edos 2, 4, 79, and 158.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[501 -316⟩	[⟨316 501]]	−0.182	0.182	4.79
2.3.5	[8 14 -13⟩, [41 -20 -4⟩	[⟨316 501 734]]	−0.269	0.193	5.08
2.3.5.7	3136/3125, 4375/4374, [-26 -1 1 9⟩	[⟨316 501 734 887]]	−0.160	0.252	6.64
2.3.5.7.11	3025/3024, 3136/3125, 4375/4374, 131072/130977	[⟨316 501 734 887 1093]]	−0.088	0.267	7.04
2.3.5.7.11.13	1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125	[⟨316 501 734 887 1093 1169]]	−0.016	0.293	7.72

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	51\316	193.67	28/25	Didacus
1	83\316	315.19	6/5	Parakleismic (7-limit)
1	84\316	322.78	3087/2560	Seniority
1	141\316	535.44	512/375	Maquila (5-limit)
1	149\316	565.82	18/13	Threedic
1	155\316	588.61	45927/32768	Countritonic (7-limit)
2	55\316	208.86	44/39	Abigail
2	83\316 (75\316)	315.19 (284.81)	6/5 (33/28)	Semiparakleismic
4	131\316 (27\316)	497.47 (102.53)	4/3 (35/33)	Unlit

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