164edo

← 163edo

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 supplies the optimal patent val for the würschmidt temperament. In higher limits, also supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit the momentous temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440.

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)
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