161edo: Difference between revisions

← 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

← Older edit

VisualWikitext

Latest revision as of 20:25, 10 April 2026

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.

Theory

161edo has a perfect fifth slightly sharp of that of 12edo, such that it maps the Pythagorean comma to one step. It approximates many of the low primes fairly well; however, it is only consistent to the 7-odd-limit, due to 10/9 being mapped too sharply from prime 5 being sharp, while 3 is flat. Nonetheless it does well for its size in higher limits, with the inconsistent intervals in the 23-odd-limit being 9/5, 13/9, 23/13, and their octave complements, and additional inconsistencies in the 25-odd-limit include 25/18, 25/23, and their octave complements. Prime 29 is also accurate, though harmonic 27 is mapped inconsistently flat, causing many of its intervals to be inconsistent. Additionally, the flatness of 27 causes 28/27 to be mapped wider than 27/26, meaning 161edo is at most diamond monotone in the 25-odd-limit.

As an equal temperament, 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

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
Error	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
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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

@@ Line 1: / Line 1: @@
-The ''161 equal division'' divides the octave into 161 equal parts of 7.453 cents each. It 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 1188/1183, 351/350, 847/845, 1575/1573, 1001/1000 and 1716/1715 in the 13-limit. It serves as the optimal patent val for [[Breedsmic_temperaments#Mintone|mintone]] temperament in the 5-, 7-, 11- and 13-limits.
+{{Infobox ET}}
+{{ED intro}}
-== Just approximation ==
+== Theory ==
-edo is notable as being low in [[29-limit]] relative error in the 100 to 200 range.
+edo has a [[perfect fifth]] slightly sharp of that of [[12edo]], such that it maps the [[Pythagorean comma]] to one step. It approximates many of the low primes fairly well; however, it is only consistent to the [[7-odd-limit]], due to [[10/9]] being mapped too sharply from prime [[5/1|5]] being sharp, while [[3/1|3]] is flat. Nonetheless it does well for its size in higher limits, with the inconsistent intervals in the [[23-odd-limit]] being 9/5, [[13/9]], [[23/13]], and their [[octave complement]]s, and additional inconsistencies in the [[25-odd-limit]] include [[25/18]], [[25/23]], and their octave complements. Prime [[29/1|29]] is also accurate, though harmonic [[27/1|27]] is mapped inconsistently flat, causing many of its intervals to be inconsistent. Additionally, the flatness of 27 causes [[28/27]] to be mapped wider than [[27/26]], meaning 161edo is at most [[diamond monotone]] in the 25-odd-limit.
-{{Primes in edo|161|columns=10}}
+As an equal temperament, 161et [[tempering out|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 ===
+{{Harmonics in equal|161}}
+=== Subsets and supersets ===
+Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.
+== Intervals ==
+{{Interval table}}
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| -255 161 }}
+| {{mapping| 161 255 }}
+| +0.421
+| 0.421
+| 5.65
+|-
+| 2.3.5
+| 393216/390625, {{monzo| -17 21 -7 }}
+| {{mapping| 161 255 374 }}
+| +0.099
+| 0.570
+| 7.65
+|-
+| 2.3.5.7
+| 2401/2400, 3136/3125, 177147/175000
+| {{mapping| 161 255 374 452 }}
+| +0.064
+| 0.498
+| 6.67
+|-
+| 2.3.5.7.11
+| 243/242, 441/440, 3136/3125, 35937/35840
+| {{mapping| 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
+| {{mapping| 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
+| {{mapping| 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
+| {{mapping| 161 255 374 452 557 596 658 684 }}
+| −0.034
+| 0.397
+| 5.32
+|}
+* 161et has a lower [[TE error|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 ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />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<br />(2\161)
+| 499.38<br />(14.91)
+| 4/3<br />(81/80)
+| [[Absurdity]]
+|}
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
+[[Category:Mintone]]

161edo: Difference between revisions

Latest revision as of 20:25, 10 April 2026

Contents

Theory

Prime harmonics

Subsets and supersets

Intervals

Regular temperament properties

Rank-2 temperaments

Navigation menu

161edo: Difference between revisions

Latest revision as of 20:25, 10 April 2026

Theory

Prime harmonics

Subsets and supersets

Intervals

Regular temperament properties

Rank-2 temperaments

Navigation menu

Search