161edo

← 160edo

161edo

162edo →

Prime factorization	7 × 23
Step size	7.45342 ¢
Fifth	94\161 (700.621 ¢)
Semitones (A1:m2)	14:13 (104.3 ¢ : 96.89 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

161et tempers out the würschmidt comma, 393216/390625, in the 5-limit; 3136/3125, 6144/6125 and 2401/2400 in the 7-limit; 243/242, 441/440, 540/539 and 5632/5625 in the 11-limit; and 351/350, 847/845, 1001/1000, 1188/1183, 1575/1573 and 1716/1715 in the 13-limit. It serves as the optimal patent val for the mintone temperament in the 5-, 7-, 11- and 13-limit.

Prime harmonics

In the range of edos from 100 to 200, 161edo is notable as being low in 29-limit relative error.

Approximation of prime harmonics in 161edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-1.33	+1.26	+0.12	+0.23	+1.71	-0.61	+0.62	-2.19	-1.01	+2.79
	Relative (%)	+0.0	-17.9	+17.0	+1.6	+3.2	+22.9	-8.2	+8.4	-29.3	-13.5	+37.4
Steps (reduced)		161 (0)	255 (94)	374 (52)	452 (130)	557 (74)	596 (113)	658 (14)	684 (40)	728 (84)	782 (138)	798 (154)

Subsets and supersets

Since 161 factors into 7 × 23, 161edo contains 7edo and 23edo as its subsets.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-255 161⟩	[⟨161 255]]	+0.421	0.421	5.65
2.3.5	393216/390625, [-17 21 -7⟩	[⟨161 255 374]]	+0.099	0.570	7.65
2.3.5.7	2401/2400, 3136/3125, 177147/175000	[⟨161 255 374 452]]	+0.064	0.498	6.67
2.3.5.7.11	243/242, 441/440, 3136/3125, 35937/35840	[⟨161 255 374 452 557]]	+0.037	0.448	6.01
2.3.5.7.11.13	243/242, 351/350, 441/440, 847/845, 3136/3125	[⟨161 255 374 452 557 596]]	−0.046	0.449	6.03
2.3.5.7.11.13.17	243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088	[⟨161 255 374 452 557 596 658]]	−0.018	0.422	5.66
2.3.5.7.11.13.17.19	243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512	[⟨161 255 374 452 557 596 658 684]]	−0.034	0.397	5.32

• 161et has a lower absolute error than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by 183edo.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	15\161	111.80	16/15	Vavoom
1	16\161	119.25	15/14	Septidiasemi
1	17\161	126.71	14/13	Mowglic
1	25\161	186.34	10/9	Mintone
1	26\161	193.79	28/25	Hemiwürschmidt
1	38\161	283.23	33/28	Neominor (161f)
1	52\161	387.58	5/4	Würschmidt (5-limit)
1	79\161	588.82	45/32	Aufo
7	67\161 (2\161)	499.38 (14.91)	4/3 (81/80)	Absurdity

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