164edo

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164edo

165edo →

Prime factorization	22 × 41
Step size	7.31707 ¢
Fifth	96\164 (702.439 ¢) (→ 24\41)
Semitones (A1:m2)	16:12 (117.1 ¢ : 87.8 ¢)
Consistency limit	5
Distinct consistency limit	5

Theory

In the 5-limit, 164edo tempers out the würschmidt comma, 393216/390625, and the vulture comma, [24 -21 4⟩. It supplies the optimal patent val for the würschmidt temperament.

In the patent val ⟨164 260 381 460 567 607], it tempers out 196/195, 352/351, 385/384, 441/440, 676/675, and supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit momentous temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440.

In the alternative val 164de ⟨164 260 381 461 568 607], it tempers out 243/242, 351/350, 364/363, 640/637, 676/675, 729/728, and 1575/1573.

164 = 4 × 41, with divisors 2, 4, 41, 82.

Prime harmonics

Approximation of prime harmonics in 164edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.48	+1.49	-2.97	-2.54	+0.94	-2.52	+2.49	+0.99	+2.13	-3.57
	Relative (%)	+0.0	+6.6	+20.4	-40.6	-34.7	+12.8	-34.4	+34.0	+13.6	+29.1	-48.8
Steps (reduced)		164 (0)	260 (96)	381 (53)	460 (132)	567 (75)	607 (115)	670 (14)	697 (41)	742 (86)	797 (141)	812 (156)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	393216/390625, [24 -21 4⟩	[⟨164 260 381]]	-0.316	0.262	3.58
2.3.5.13	676/675, 256000/255879, 393216/390625	[⟨164 260 381 607]]	-0.300	0.229	3.13

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per Otave	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	47\164	343.90	8000/6561	Geb
1	49\164	358.54	16/13	Restles (164)
1	53\164	387.80	5/4	Würschmidt
1	53\164	475.61	320/243	Vulture
1	69\164	504.88	104976/78125	Countermeantone
2	17\164	124.39	275/256	Semivulture (164)
2	25\164	182.93	10/9	Unidecmic
4	68\164 (14\164)	497.56 (102.44)	4/3 (35/33)	Undim (164deff) / unlit (164f)
41	53\164 (1\164)	387.80 (7.32)	5/4 (32805/32768)	Counterpyth