231edo

The 231 equal temperament divides the octave into 231 equal parts of 5.195 cents each.

Theory

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)

In the 5-limit it tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it supports 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 - see here.

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, 14700/14641, 2460375/2458624	[⟨231 366 536 648 799]]	0.469	0.354	6.81

Rank two temperaments by generator

Periods per octave	Generator	Cents	Associated ratio	Temperaments
1	62\231	322.08		Dee leap week
3	61\231 (16\231)	316.88 (83.12)	6/5	Tritrikleismic

References

https://individual.utoronto.ca/kalendis/leap/index.htm