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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro}}
{{ED intro}}


== Theory ==
== Theory ==
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.
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 [[5/1|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.
 
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 ===
=== Prime harmonics ===
In the range of edos from 100 to 200, 161edo is notable as being low in [[29-limit]] relative error.


{{Harmonics in equal|161}}
{{Harmonics in equal|161}}
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.
Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.
== Intervals ==
{{Interval table}}


== Regular temperament properties ==
== Regular temperament properties ==
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| [[Absurdity]]
| [[Absurdity]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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]]
[[Category:Mintone]]

Latest revision as of 20:25, 10 April 2026

← 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

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 21/161, 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
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

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