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.

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)

Subsets and supersets

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.

Regular temperament properties

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 26\231 | 135.06 | 27/25 | Superlimmal |- | 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 Template:Rank-2 end Template:Orf

Music

Mercury Amalgam

• Sins of Stoicism (Demo Version, March 2022)