405edo

Theory

405et is consistent to the 7-odd-limit. Using the patent val, it tempers out 4096000/4084101, 2460375/2458624, 33554432/33480783, 2100875/2097152 and 40353607/40310784 in the 7-limit; 56723625/56689952, 55296000/55240493, 117440512/117406179, 820125/819896, 4000/3993, 78675968/78594219, 7168000/7144929, 496125/495616, 117649/117612, 1296000/1294139, 131072/130977, 1265625/1261568, 42875/42768, 5767168/5764801, 1375/1372, 19712/19683, 41503/41472, 199297406/199290375, 1362944/1361367, 166375/165888 and 67110351/67108864 in the 11-limit. It supports marthirds, novemkleismic and kleirtismic. It provides the optimal patent val for novemkleismic.

Approximation of prime harmonics in 405edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.27	-1.13	+0.06	-0.21	+0.95	-1.25	-1.22	-0.13	-1.43	-1.33
Error	Relative (%)	+0.0	+9.0	-38.1	+2.1	-7.0	+32.2	-42.2	-41.1	-4.3	-48.2	-45.0
Steps (reduced)		405 (0)	642 (237)	940 (130)	1137 (327)	1401 (186)	1499 (284)	1655 (35)	1720 (100)	1832 (212)	1967 (347)	2006 (386)

405 factors into 3⁴ × 5, with subset edos 3, 5, 9, 15, 27, 45, 81, and 135.

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3	[214 -135⟩	[⟨405 642]]	-0.0843	0.0843	2.85
2.3.5	15625/15552, [110 -65 -3⟩	[⟨405 642 940]]	+0.1058	0.2776	9.37
2.3.5.7	15625/15552, 2460375/2458624, 4096000/4084101	[⟨405 642 940 1137]]	+0.0737	0.2467	8.33
2.3.5.7.11	1375/1372, 42875/42768, 19712/19683, 41503/41472	[⟨405 642 940 1137 1401]]	+0.0709	0.2207	7.45

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	56\405	165.93	8575/7776	Satin
1	107\405	317.04	6/5	Hanson
9	107\405 (17\405)	317.04 (50.37)	6/5 (36/35)	Novemkleismic

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