161edo

161 equal divisions of the octave (abbreviated 161edo or 161ed2), also called 161-tone equal temperament (161tet) or 161 equal temperament (161et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 161 equal parts of about 7.45 ¢ each. Each step represents a frequency ratio of 2^1/161, or the 161st root of 2.

← 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.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	7.5		^D, ^^E♭♭
2	14.9		^^D, ^3E♭♭
3	22.4		^3D, ^4E♭♭
4	29.8	56/55, 57/56, 58/57	^4D, ^5E♭♭
5	37.3	46/45, 47/46, 48/47	^5D, ^6E♭♭
6	44.7	39/38, 40/39	^6D, v7E♭
7	52.2	33/32, 34/33	^7D, v6E♭
8	59.6	29/28, 30/29	v6D♯, v5E♭
9	67.1	26/25	v5D♯, v4E♭
10	74.5	24/23, 47/45	v4D♯, v3E♭
11	82	43/41	v3D♯, vvE♭
12	89.4	20/19	vvD♯, vE♭
13	96.9	37/35, 55/52	vD♯, E♭
14	104.3	17/16	D♯, ^E♭
15	111.8	16/15	^D♯, ^^E♭
16	119.3	15/14	^^D♯, ^3E♭
17	126.7		^3D♯, ^4E♭
18	134.2	40/37	^4D♯, ^5E♭
19	141.6	38/35, 51/47	^5D♯, ^6E♭
20	149.1		^6D♯, v7E
21	156.5	23/21, 35/32	^7D♯, v6E
22	164	11/10	v6D𝄪, v5E
23	171.4	32/29	v5D𝄪, v4E
24	178.9	41/37, 51/46	v4D𝄪, v3E
25	186.3	39/35, 49/44	v3D𝄪, vvE
26	193.8	19/17, 47/42	vvD𝄪, vE
27	201.2	55/49	E
28	208.7	35/31, 44/39	^E, ^^F♭
29	216.1	17/15	^^E, ^3F♭
30	223.6	33/29, 58/51	^3E, ^4F♭
31	231.1	8/7	^4E, ^5F♭
32	238.5	39/34	^5E, ^6F♭
33	246		^6E, v7F
34	253.4	22/19	^7E, v6F
35	260.9	43/37, 50/43, 57/49	v6E♯, v5F
36	268.3		v5E♯, v4F
37	275.8	34/29	v4E♯, v3F
38	283.2	33/28	v3E♯, vvF
39	290.7	58/49	vvE♯, vF
40	298.1	19/16	F
41	305.6	31/26, 37/31	^F, ^^G♭♭
42	313		^^F, ^3G♭♭
43	320.5		^3F, ^4G♭♭
44	328	29/24, 52/43	^4F, ^5G♭♭
45	335.4	17/14	^5F, ^6G♭♭
46	342.9	39/32, 50/41	^6F, v7G♭
47	350.3	49/40, 60/49	^7F, v6G♭
48	357.8	43/35	v6F♯, v5G♭
49	365.2	21/17, 58/47	v5F♯, v4G♭
50	372.7	31/25	v4F♯, v3G♭
51	380.1		v3F♯, vvG♭
52	387.6	5/4	vvF♯, vG♭
53	395	44/35, 49/39	vF♯, G♭
54	402.5	29/23	F♯, ^G♭
55	409.9	19/15	^F♯, ^^G♭
56	417.4	14/11	^^F♯, ^3G♭
57	424.8	23/18, 55/43	^3F♯, ^4G♭
58	432.3		^4F♯, ^5G♭
59	439.8	49/38, 58/45	^5F♯, ^6G♭
60	447.2	22/17, 57/44	^6F♯, v7G
61	454.7	13/10	^7F♯, v6G
62	462.1	47/36	v6F𝄪, v5G
63	469.6	21/16	v5F𝄪, v4G
64	477	29/22	v4F𝄪, v3G
65	484.5	41/31, 45/34	v3F𝄪, vvG
66	491.9		vvF𝄪, vG
67	499.4	4/3	G
68	506.8		^G, ^^A♭♭
69	514.3	35/26, 39/29	^^G, ^3A♭♭
70	521.7	50/37	^3G, ^4A♭♭
71	529.2	19/14	^4G, ^5A♭♭
72	536.6	15/11	^5G, ^6A♭♭
73	544.1	26/19	^6G, v7A♭
74	551.6	11/8	^7G, v6A♭
75	559	29/21	v6G♯, v5A♭
76	566.5	43/31	v5G♯, v4A♭
77	573.9	39/28, 46/33	v4G♯, v3A♭
78	581.4	7/5	v3G♯, vvA♭
79	588.8	45/32, 52/37	vvG♯, vA♭
80	596.3	24/17, 55/39	vG♯, A♭
81	603.7	17/12	G♯, ^A♭
82	611.2	37/26, 47/33	^G♯, ^^A♭
83	618.6	10/7	^^G♯, ^3A♭
84	626.1	33/23, 56/39	^3G♯, ^4A♭
85	633.5	49/34	^4G♯, ^5A♭
86	641	42/29, 55/38	^5G♯, ^6A♭
87	648.4	16/11	^6G♯, v7A
88	655.9	19/13	^7G♯, v6A
89	663.4	22/15	v6G𝄪, v5A
90	670.8	28/19	v5G𝄪, v4A
91	678.3	37/25	v4G𝄪, v3A
92	685.7	49/33, 52/35, 55/37, 58/39	v3G𝄪, vvA
93	693.2		vvG𝄪, vA
94	700.6	3/2	A
95	708.1		^A, ^^B♭♭
96	715.5		^^A, ^3B♭♭
97	723	44/29	^3A, ^4B♭♭
98	730.4	32/21	^4A, ^5B♭♭
99	737.9	49/32	^5A, ^6B♭♭
100	745.3	20/13	^6A, v7B♭
101	752.8	17/11	^7A, v6B♭
102	760.2	45/29	v6A♯, v5B♭
103	767.7		v5A♯, v4B♭
104	775.2	36/23	v4A♯, v3B♭
105	782.6	11/7	v3A♯, vvB♭
106	790.1	30/19	vvA♯, vB♭
107	797.5	46/29	vA♯, B♭
108	805	35/22	A♯, ^B♭
109	812.4	8/5	^A♯, ^^B♭
110	819.9		^^A♯, ^3B♭
111	827.3	50/31	^3A♯, ^4B♭
112	834.8	34/21, 47/29	^4A♯, ^5B♭
113	842.2		^5A♯, ^6B♭
114	849.7	49/30	^6A♯, v7B
115	857.1	41/25	^7A♯, v6B
116	864.6	28/17	v6A𝄪, v5B
117	872	43/26, 48/29	v5A𝄪, v4B
118	879.5		v4A𝄪, v3B
119	887		v3A𝄪, vvB
120	894.4	52/31, 57/34	vvA𝄪, vB
121	901.9	32/19	B
122	909.3	49/29	^B, ^^C♭
123	916.8	56/33	^^B, ^3C♭
124	924.2	29/17	^3B, ^4C♭
125	931.7		^4B, ^5C♭
126	939.1	43/25	^5B, ^6C♭
127	946.6	19/11	^6B, v7C
128	954		^7B, v6C
129	961.5		v6B♯, v5C
130	968.9	7/4	v5B♯, v4C
131	976.4	51/29, 58/33	v4B♯, v3C
132	983.9	30/17	v3B♯, vvC
133	991.3	39/22, 55/31	vvB♯, vC
134	998.8	57/32	C
135	1006.2	34/19	^C, ^^D♭♭
136	1013.7		^^C, ^3D♭♭
137	1021.1		^3C, ^4D♭♭
138	1028.6	29/16	^4C, ^5D♭♭
139	1036	20/11	^5C, ^6D♭♭
140	1043.5	42/23	^6C, v7D♭
141	1050.9		^7C, v6D♭
142	1058.4	35/19	v6C♯, v5D♭
143	1065.8	37/20	v5C♯, v4D♭
144	1073.3		v4C♯, v3D♭
145	1080.7	28/15	v3C♯, vvD♭
146	1088.2	15/8	vvC♯, vD♭
147	1095.7	32/17	vC♯, D♭
148	1103.1		C♯, ^D♭
149	1110.6	19/10	^C♯, ^^D♭
150	1118		^^C♯, ^3D♭
151	1125.5	23/12	^3C♯, ^4D♭
152	1132.9	25/13	^4C♯, ^5D♭
153	1140.4	29/15, 56/29	^5C♯, ^6D♭
154	1147.8	33/17	^6C♯, v7D
155	1155.3	39/20	^7C♯, v6D
156	1162.7	45/23, 47/24	v6C𝄪, v5D
157	1170.2	55/28, 57/29	v5C𝄪, v4D
158	1177.6		v4C𝄪, v3D
159	1185.1		v3C𝄪, vvD
160	1192.5		vvC𝄪, vD
161	1200	2/1	D

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