210edo

← 209edo

210edo

211edo →

Prime factorization	2 × 3 × 5 × 7
Step size	5.71429 ¢
Fifth	123\210 (702.857 ¢) (→ 41\70)
Semitones (A1:m2)	21:15 (120 ¢ : 85.71 ¢)
Consistency limit	9
Distinct consistency limit	9

Theory

210et tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit. It is consistent to the 7-limit, but there is a sharp tendency for harmonics 3, 5, and 7. Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 210edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.90	+2.26	+2.60	+1.80	-2.75	-0.53	-2.55	-2.10	-0.37	-2.21	+0.30
	Relative (%)	+15.8	+39.5	+45.5	+31.6	-48.1	-9.2	-44.7	-36.7	-6.5	-38.7	+5.2
Steps (reduced)		333 (123)	488 (68)	590 (170)	666 (36)	726 (96)	777 (147)	820 (190)	858 (18)	892 (52)	922 (82)	950 (110)

Subsets and supersets

210edo factors into 2 × 3 × 5 × 7, with subset edos 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[111 -70⟩	⟨210 333]	-0.2846	0.2845	4.98
2.3.5	[26 -12 -3⟩, [19 10 -15⟩	⟨210 333 488]	-0.5138	0.3987	6.98
2.3.5.7	3136/3125, 5120/5103, 118098/117649	⟨210 333 488 590]	-0.6170	0.3888	6.80

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
3	123\210 (17\210)	702.86 (97.14)	3/2 (135\128)	Misty
5	98\210 (13\210)	560.00 (74.29)	864/625 (25/24)	Trisedodge