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

164 = 4 × 41, and 164edo shares its fifth with 41edo. In the 5-limit, 164et 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. The 164dg val is a good tuning for 7- to 19-limit buzzard temperament, although if harmonic 11 is desired it is only easily accessible through the patent mapping.

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)

Subsets and supersets

Since 164 = 2² × 41, 164edo has subset edos 2, 4, 41, 82. 328edo, which doubles it, provides good correction for the approximation to harmonics 7 and 11, and is consistent in the 13-odd-limit.

Regular temperament properties

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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) | Countercomp Template:Rank-2 end Template:Orf