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181edo

Revision as of 05:18, 21 February 2026 by Overthink (talk | contribs) (→Intervals: substitute)

181 equal divisions of the octave (abbreviated 181edo or 181ed2), also called 181-tone equal temperament (181tet) or 181 equal temperament (181et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 181 equal parts of about 6.63 ¢ each. Each step represents a frequency ratio of 21/181, or the 181st root of 2.

← 180edo 181edo 182edo →
Prime factorization 181 (prime)
Step size 6.62983 ¢ 
Fifth 106\181 (702.762 ¢)
Semitones (A1:m2) 18:13 (119.3 ¢ : 86.19 ¢)
Consistency limit 7
Distinct consistency limit 7

Theory

181edo is only consistent to the 7-odd-limit, though except for 9/5, 23/20 and their octave complements, it is consistent to the 23-odd-limit. Beyond that, it does well on primes 37 and 43, and has unambiguous though not accurate approximations to 29, 31, and 41. However, the composite harmonics 25, 27, 35, and 39 cause inconsistencies, with harmonic 25 itself being inconsistent.

As an equal temperament, 181et tempers out 2109375/2097152 (semicomma) and [14 -22 9⟩ in the 5-limit; 2401/2400, 5120/5103, and 390625/387072 in the 7-limit (supporting the hemififths and the cotritone). Using the patent val, it tempers out 385/384, 1375/1372, 2200/2187, and 4000/3993 in the 11-limit; and 325/324, 352/351, 847/845, and 1575/1573 in the 13-limit. It tempers out 375/374, 595/594, and 1275/1274 in the 17-limit, 400/399 in the 19-limit, and 300/299 in the 23-limit.

Because its harmonic 5 causes some inconsistencies, and is less accurate than the other harmonics, 181edo can reasonably be treated as a no-5 system, where it is purely consistent[idiosyncratic term] (meaning all harmonics have under 25% relative error) up to the 23-odd-limit. It tempers out [15 -13 2⟩ and [-31 -7 15⟩ in the 2.3.7 subgroup; 26411/26244, 43923/43904, and 131072/130977 in the 2.3.7.11 subgroup; and 352/351, 20449/20412, 31213/31104, and 53361/53248 in the 2.3.7.11.13 subgroup. It tempers out 833/832 and 1089/1088 in the no-5 17-limit, 343/342, 1729/1728, and 2432/2431 in the no-5 19-limit, and 392/391 in the no-5 23-limit.

Prime harmonics

Approximation of prime harmonics in 181edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.81 -1.78 -0.87 -1.04 +1.46 +1.12 +0.83 +1.56 -1.95 +1.93
Relative (%) +0.0 +12.2 -26.9 -13.1 -15.7 +22.0 +16.9 +12.5 +23.5 -29.5 +29.0
Steps
(reduced) 		181
(0) 		287
(106) 		420
(58) 		508
(146) 		626
(83) 		670
(127) 		740
(16) 		769
(45) 		819
(95) 		879
(155) 		897
(173)
Approximation of prime harmonics in 181edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +0.59 +1.88 -1.02 -2.52 +1.63 +1.60 -3.07 +0.25 -0.69 -2.38 +0.10
Relative (%) +8.9 +28.3 -15.4 -38.1 +24.6 +24.2 -46.3 +3.8 -10.4 -35.8 +1.6
Steps
(reduced) 		943
(38) 		970
(65) 		982
(77) 		1005
(100) 		1037
(132) 		1065
(160) 		1073
(168) 		1098
(12) 		1113
(27) 		1120
(34) 		1141
(55)

Subsets and supersets

181edo is the 42nd prime edo.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 6.63 ^D, ^6E♭♭
2 13.26 ^^D, ^7E♭♭
3 19.89 ^3D, ^8E♭♭
4 26.52 63/62, 64/63, 65/64 ^4D, v9E♭
5 33.15 52/51 ^5D, v8E♭
6 39.78 43/42, 44/43, 45/44 ^6D, v7E♭
7 46.41 37/36, 38/37 ^7D, v6E♭
8 53.04 33/32, 65/63 ^8D, v5E♭
9 59.67 29/28, 30/29 ^9D, v4E♭
10 66.3 27/26 v8D♯, v3E♭
11 72.93 24/23, 49/47 v7D♯, vvE♭
12 79.56 22/21, 45/43 v6D♯, vE♭
13 86.19 41/39 v5D♯, E♭
14 92.82 19/18, 58/55 v4D♯, ^E♭
15 99.45 18/17 v3D♯, ^^E♭
16 106.08 17/16, 50/47 vvD♯, ^3E♭
17 112.71 16/15 vD♯, ^4E♭
18 119.34 15/14 D♯, ^5E♭
19 125.97 43/40 ^D♯, ^6E♭
20 132.6 41/38 ^^D♯, ^7E♭
21 139.23 13/12 ^3D♯, ^8E♭
22 145.86 37/34, 62/57 ^4D♯, v9E
23 152.49 ^5D♯, v8E
24 159.12 34/31, 57/52 ^6D♯, v7E
25 165.75 11/10 ^7D♯, v6E
26 172.38 21/19 ^8D♯, v5E
27 179.01 51/46 ^9D♯, v4E
28 185.64 49/44 v8D𝄪, v3E
29 192.27 19/17 v7D𝄪, vvE
30 198.9 37/33, 46/41, 55/49 v6D𝄪, vE
31 205.52 E
32 212.15 26/23 ^E, ^6F♭
33 218.78 42/37 ^^E, ^7F♭
34 225.41 41/36, 49/43 ^3E, ^8F♭
35 232.04 8/7 ^4E, v9F
36 238.67 31/27, 39/34 ^5E, v8F
37 245.3 38/33 ^6E, v7F
38 251.93 37/32 ^7E, v6F
39 258.56 36/31, 65/56 ^8E, v5F
40 265.19 ^9E, v4F
41 271.82 48/41, 55/47 v8E♯, v3F
42 278.45 27/23, 47/40 v7E♯, vvF
43 285.08 33/28, 46/39 v6E♯, vF
44 291.71 45/38, 58/49 F
45 298.34 19/16 ^F, ^6G♭♭
46 304.97 31/26 ^^F, ^7G♭♭
47 311.6 ^3F, ^8G♭♭
48 318.23 ^4F, v9G♭
49 324.86 35/29, 41/34 ^5F, v8G♭
50 331.49 23/19, 63/52 ^6F, v7G♭
51 338.12 45/37, 62/51 ^7F, v6G♭
52 344.75 ^8F, v5G♭
53 351.38 38/31, 49/40, 60/49 ^9F, v4G♭
54 358.01 v8F♯, v3G♭
55 364.64 21/17, 58/47 v7F♯, vvG♭
56 371.27 57/46 v6F♯, vG♭
57 377.9 46/37, 51/41, 56/45 v5F♯, G♭
58 384.53 v4F♯, ^G♭
59 391.16 v3F♯, ^^G♭
60 397.79 39/31 vvF♯, ^3G♭
61 404.42 24/19 vF♯, ^4G♭
62 411.05 52/41 F♯, ^5G♭
63 417.68 14/11 ^F♯, ^6G♭
64 424.31 23/18 ^^F♯, ^7G♭
65 430.94 ^3F♯, ^8G♭
66 437.57 ^4F♯, v9G
67 444.2 31/24 ^5F♯, v8G
68 450.83 48/37 ^6F♯, v7G
69 457.46 43/33, 56/43 ^7F♯, v6G
70 464.09 17/13 ^8F♯, v5G
71 470.72 21/16 ^9F♯, v4G
72 477.35 29/22, 54/41 v8F𝄪, v3G
73 483.98 41/31 v7F𝄪, vvG
74 490.61 v6F𝄪, vG
75 497.24 4/3 G
76 503.87 ^G, ^6A♭♭
77 510.5 43/32, 47/35, 51/38 ^^G, ^7A♭♭
78 517.13 31/23, 58/43 ^3G, ^8A♭♭
79 523.76 23/17, 65/48 ^4G, v9A♭
80 530.39 ^5G, v8A♭
81 537.02 15/11 ^6G, v7A♭
82 543.65 26/19, 63/46 ^7G, v6A♭
83 550.28 11/8 ^8G, v5A♭
84 556.91 40/29 ^9G, v4A♭
85 563.54 18/13 v8G♯, v3A♭
86 570.17 57/41 v7G♯, vvA♭
87 576.8 60/43 v6G♯, vA♭
88 583.43 7/5 v5G♯, A♭
89 590.06 45/32, 52/37 v4G♯, ^A♭
90 596.69 24/17 v3G♯, ^^A♭
91 603.31 17/12 vvG♯, ^3A♭
92 609.94 37/26, 64/45 vG♯, ^4A♭
93 616.57 10/7 G♯, ^5A♭
94 623.2 43/30 ^G♯, ^6A♭
95 629.83 ^^G♯, ^7A♭
96 636.46 13/9 ^3G♯, ^8A♭
97 643.09 29/20 ^4G♯, v9A
98 649.72 16/11 ^5G♯, v8A
99 656.35 19/13 ^6G♯, v7A
100 662.98 22/15 ^7G♯, v6A
101 669.61 ^8G♯, v5A
102 676.24 34/23, 65/44 ^9G♯, v4A
103 682.87 43/29, 46/31 v8G𝄪, v3A
104 689.5 64/43 v7G𝄪, vvA
105 696.13 v6G𝄪, vA
106 702.76 3/2 A
107 709.39 ^A, ^6B♭♭
108 716.02 62/41, 65/43 ^^A, ^7B♭♭
109 722.65 41/27, 44/29 ^3A, ^8B♭♭
110 729.28 32/21 ^4A, v9B♭
111 735.91 26/17 ^5A, v8B♭
112 742.54 43/28, 63/41 ^6A, v7B♭
113 749.17 37/24, 57/37 ^7A, v6B♭
114 755.8 48/31, 65/42 ^8A, v5B♭
115 762.43 ^9A, v4B♭
116 769.06 v8A♯, v3B♭
117 775.69 36/23 v7A♯, vvB♭
118 782.32 11/7 v6A♯, vB♭
119 788.95 41/26 v5A♯, B♭
120 795.58 19/12 v4A♯, ^B♭
121 802.21 62/39 v3A♯, ^^B♭
122 808.84 vvA♯, ^3B♭
123 815.47 vA♯, ^4B♭
124 822.1 37/23, 45/28 A♯, ^5B♭
125 828.73 ^A♯, ^6B♭
126 835.36 34/21, 47/29 ^^A♯, ^7B♭
127 841.99 ^3A♯, ^8B♭
128 848.62 31/19, 49/30 ^4A♯, v9B
129 855.25 ^5A♯, v8B
130 861.88 51/31 ^6A♯, v7B
131 868.51 38/23 ^7A♯, v6B
132 875.14 58/35, 63/38 ^8A♯, v5B
133 881.77 ^9A♯, v4B
134 888.4 v8A𝄪, v3B
135 895.03 52/31, 57/34 v7A𝄪, vvB
136 901.66 32/19 v6A𝄪, vB
137 908.29 49/29 B
138 914.92 39/23, 56/33 ^B, ^6C♭
139 921.55 46/27, 63/37 ^^B, ^7C♭
140 928.18 41/24, 65/38 ^3B, ^8C♭
141 934.81 ^4B, v9C
142 941.44 31/18 ^5B, v8C
143 948.07 64/37 ^6B, v7C
144 954.7 33/19 ^7B, v6C
145 961.33 54/31 ^8B, v5C
146 967.96 7/4 ^9B, v4C
147 974.59 65/37 v8B♯, v3C
148 981.22 37/21 v7B♯, vvC
149 987.85 23/13 v6B♯, vC
150 994.48 C
151 1001.1 41/23 ^C, ^6D♭♭
152 1007.73 34/19 ^^C, ^7D♭♭
153 1014.36 ^3C, ^8D♭♭
154 1020.99 ^4C, v9D♭
155 1027.62 38/21 ^5C, v8D♭
156 1034.25 20/11 ^6C, v7D♭
157 1040.88 31/17 ^7C, v6D♭
158 1047.51 ^8C, v5D♭
159 1054.14 57/31 ^9C, v4D♭
160 1060.77 24/13 v8C♯, v3D♭
161 1067.4 63/34 v7C♯, vvD♭
162 1074.03 v6C♯, vD♭
163 1080.66 28/15 v5C♯, D♭
164 1087.29 15/8 v4C♯, ^D♭
165 1093.92 32/17, 47/25 v3C♯, ^^D♭
166 1100.55 17/9 vvC♯, ^3D♭
167 1107.18 36/19, 55/29 vC♯, ^4D♭
168 1113.81 C♯, ^5D♭
169 1120.44 21/11 ^C♯, ^6D♭
170 1127.07 23/12 ^^C♯, ^7D♭
171 1133.7 52/27 ^3C♯, ^8D♭
172 1140.33 29/15, 56/29 ^4C♯, v9D
173 1146.96 64/33 ^5C♯, v8D
174 1153.59 37/19 ^6C♯, v7D
175 1160.22 43/22 ^7C♯, v6D
176 1166.85 51/26 ^8C♯, v5D
177 1173.48 63/32, 65/33 ^9C♯, v4D
178 1180.11 v8C𝄪, v3D
179 1186.74 v7C𝄪, vvD
180 1193.37 v6C𝄪, vD
181 1200 2/1 D

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [287 -181⟩ [⟨181 287]] −0.255 0.255 3.84
2.3.5 2109375/2097152, [14 -22 9⟩ [⟨181 287 420]] +0.086 0.525 7.92
2.3.5.7 2401/2400, 5120/5103, 390625/387072 [⟨181 287 420 508]] +0.142 0.465 7.01
2.3.5.7.11 385/384, 1375/1372, 2200/2187, 4000/3993 [⟨181 287 420 508 626]] +0.174 0.421 6.35
2.3.5.7.11.13 325/324, 352/351, 385/384, 1375/1372, 1575/1573 [⟨181 287 420 508 626 670]] +0.079 0.439 6.62
2.3.5.7.11.13.17 325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274 [⟨181 287 420 508 626 670 740]] +0.028 0.425 6.40
2.3.5.7.11.13.17.19 325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274 [⟨181 287 420 508 626 670 740 769]] +0.000 0.404 6.09

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 18\181 119.34 15/14 Septidiasemi
1 35\181 232.04 8/7 Quadrawell
1 39\181 258.56 [-32 13 5⟩ Lafa
1 41\181 271.82 75/64 Orson
1 53\181 351.38 49/40 Hemififths (7-limit)
1 88\181 583.43 7/5 Cotritone (181f)

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

Music

Francium

See also

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