231edo

← 230edo

231edo

232edo →

Prime factorization	3 × 7 × 11
Step size	5.19481 ¢
Fifth	135\231 (701.299 ¢) (→ 45\77)
Semitones (A1:m2)	21:18 (109.1 ¢ : 93.51 ¢)
Consistency limit	11
Distinct consistency limit	11

Theory

In the 5-limit, 231et tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it supports the tritikleismic temperament, and in fact provides the optimal patent val. In the 11-limit it tempers out 385/384, 441/440 and 4000/3993, leading to 11-limit tritikleismic for which it also gives the optimal patent val.

231 years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a 41 & 149 temperament tempering out 132055/131072, 166375/165888, and 2460375/2458624. This type of solar calendar leap rule scale may actually be of more use to harmony, since a 41 note subset mimics 41edo, a rather useful edo harmonically, and it preserves the simple commas mentioned above.

231 = 3 × 7 × 11, with subset edos 3, 7, 11, 21, 33, and 77. Since it contains 77edo, it can be used for playing such a tuning of the Carlos Alpha scale.

Odd harmonics

Approximation of odd harmonics in 231edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-0.66	-1.90	-2.59	-1.31	-0.67	+1.03	-2.55	-1.06	-1.41	+1.95	+0.30
	Relative (%)	-12.6	-36.5	-49.9	-25.3	-12.9	+19.8	-49.2	-20.4	-27.1	+37.5	+5.7
Steps (reduced)		366 (135)	536 (74)	648 (186)	732 (39)	799 (106)	855 (162)	902 (209)	944 (20)	981 (57)	1015 (91)	1045 (121)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	15625/15552, [-64 36 3⟩	[⟨231 366 536]]	0.410	0.334	6.43
2.3.5.7	1029/1024, 15625/15552, 823543/820125	[⟨231 366 536 648]]	0.539	0.365	7.01
2.3.5.7.11	385/384, 441/440, 4000/3993, 823543/820125	[⟨231 366 536 648 799]]	0.469	0.354	6.81

Rank-2 temperaments

Periods per octave	Generator	Cents	Associated ratio	Temperaments
1	27\231	140.26	243/224	Septichrome
1	45\231	233.77	8/7	Slendric
1	61\231	316.88	6/5	Hanson
1	62\231	322.08	135/112	Dee leap week
1	73\231	379.22	56/45	Marthirds
3	61\231 (16\231)	316.88 (83.12)	6/5 (21/20)	Tritikleismic