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'''181edo''' is the [[EDO|equal division of the octave]] into 181 parts of 6.6298 cents each. It tempers out 2109375/2097152 (semicomma) and 32000000000/31381059609 in the 5-limit; 2401/2400, 5120/5103, and 390625/387072 in the 7-limit (supporting the [[Breedsmic temperaments|hemififths]] and the [[Breedsmic temperaments|cotritone]]). Using the patent val, it tempers out 385/384, 1375/1372, 2200/2197, and 4000/3993 in the 11-limit; 325/324, 352/351, 847/845, and 1575/1573 in the 13-limit.
{{Infobox ET}}
{{ED intro}}


181edo is the 42nd [[prime EDO]].
== Theory ==
181edo is only [[consistent]] to the [[7-odd-limit]], though except for [[9/5]], [[23/20]] and their [[octave complement]]s, it is consistent to the [[23-odd-limit]]. Beyond that, it does well on [[prime interval|primes]] [[37/1|37]] and [[43/1|43]], and has unambiguous though not accurate approximations to [[29/1|29]], [[31/1|31]], and [[41/1|41]]. However, the composite harmonics [[25/1|25]], [[27/1|27]], [[35/1|35]], and [[39/1|39]] cause inconsistencies, with harmonic 25 itself being inconsistent.


[[Category:Edo]]
As an equal temperament, 181et [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo| 14 -22 9 }} in the [[5-limit]]; [[2401/2400]], [[5120/5103]], and 390625/387072 in the [[7-limit]] ([[support]]ing 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]].
[[Category:Prime EDO]]
 
Because its harmonic [[5/1|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]]{{idio}} (meaning all harmonics have under 25% [[relative error]]) up to the 23-odd-limit. It tempers out {{Monzo|15 -13 2}} and {{Monzo|-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 ===
{{Harmonics in equal|181|columns=11}}
{{Harmonics in equal|181|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 181edo (continued)}}
 
=== Subsets and supersets ===
181edo is the 42nd [[prime edo]].
 
== Intervals ==
 
{{Todo|complete table|inline=1}}
 
{| class="wikitable center-1 right-2 mw-collapsible mw-collapsed"
|-
! Steps
! Cents
! Approximate ratios*
|-
| 0
| 0
| [[1/1]]
|-
| 1
| 6.63
|
|-
| 2
| 13.26
|
|-
| 3
| 19.89
|
|-
| 4
| 26.52
| [[64/63]], [[65/64]], [[66/65]]
|-
| 5
| 33.15
| [[49/48]], [[50/49]], [[52/51]]
|-
| 6
| 39.78
| [[45/44]], ''[[51/50]]''
|-
| 7
| 46.41
| ''[[35/34]]''
|-
| 8
| 53.04
| [[33/32]], [[34/33]], ''[[36/35]]''
|-
| 9
| 59.67
|
|-
| 10
| 66.3
| ''[[25/24]],'' [[27/26]]
|-
| 11
| 72.93
| [[24/23]], ''[[26/25]]''
|-
| 12
| 79.56
| [[22/21]], [[23/22]]
|-
| 13
| 86.19
| [[20/19]], [[21/20]]
|-
| 14
| 92.82
| [[19/18]]
|-
| 15
| 99.45
| [[18/17]]
|-
| 16
| 106.08
| [[17/16]]
|-
| 17
| 112.71
| [[16/15]]
|-
| 18
| 119.34
| [[15/14]]
|-
| 19
| 125.97
| [[14/13]]
|-
| 20
| 132.6
|
|-
| 21
| 139.23
| [[13/12]], ''[[25/23]]'', ''[[27/25]]''
|-
| 22
| 145.86
|
|-
| 23
| 152.49
| [[12/11]]
|-
| 24
| 159.12
| [[23/21]]
|-
| 25
| 165.75
| [[11/10]]
|-
| 26
| 172.38
| [[21/19]]
|-
| 27
| 179.01
| ''[[10/9]],'' [[51/46]]
|-
| 28
| 185.64
| [[49/44]]
|-
| 29
| 192.27
| [[19/17]]
|-
| 30
| 198.9
| [[55/49]]
|-
| 31
| 205.52
| [[9/8]]
|-
| 32
| 212.15
| [[26/23]]
|-
| 33
| 218.78
| [[17/15]]
|-
| 34
| 225.41
|
|-
| 35
| 232.04
| [[8/7]]
|-
| 36
| 238.67
| [[39/34]]
|-
| 37
| 245.3
| [[15/13]], ''[[23/20]]'', [[38/33]]
|-
| 38
| 251.93
|
|-
| 39
| 258.56
| [[65/56]]
|-
| 40
| 265.19
|
|-
| 41
| 271.82
|
|-
| 42
| 278.45
| [[27/23]]
|-
| 43
| 285.08
| [[33/28]], [[46/39]]
|-
| 44
| 291.71
| [[45/38]]
|-
| 45
| 298.34
| [[19/16]]
|-
| 46
| 304.97
|
|-
| 47
| 311.6
|
|-
| 48
| 318.23
| [[6/5]]
|-
| 49
| 324.86
|
|-
| 50
| 331.49
| [[23/19]], [[63/52]]
|-
| 51
| 338.12
| [[62/51]]
|-
| 52
| 344.75
|
|-
| 53
| 351.38
| [[49/40]], [[60/49]]
|-
| 54
| 358.01
|
|-
| 55
| 364.64
| [[21/17]]
|-
| 56
| 371.27
| [[57/46]]
|-
| 57
| 377.9
| [[56/45]]
|-
| 58
| 384.53
| [[5/4]]
|-
| 59
| 391.16
|
|-
| 60
| 397.79
|
|-
| 61
| 404.42
| [[24/19]]
|-
| 62
| 411.05
|
|-
| 63
| 417.68
| [[14/11]]
|-
| 64
| 424.31
| [[23/18]]
|-
| 65
| 430.94
|
|-
| 66
| 437.57
|
|-
| 67
| 444.2
|
|-
| 68
| 450.83
|
|-
| 69
| 457.46
|
|-
| 70
| 464.09
| [[17/13]]
|-
| 71
| 470.72
| [[21/16]]
|-
| 72
| 477.35
|
|-
| 73
| 483.98
|
|-
| 74
| 490.61
|
|-
| 75
| 497.24
| [[4/3]]
|-
| 76
| 503.87
|
|-
| 77
| 510.5
| [[51/38]]
|-
| 78
| 517.13
|
|-
| 79
| 523.76
| [[23/17]], [[65/48]]
|-
| 80
| 530.39
|
|-
| 81
| 537.02
| [[15/11]]
|-
| 82
| 543.65
| [[26/19]], [[63/46]]
|-
| 83
| 550.28
| [[11/8]]
|-
| 84
| 556.91
|
|-
| 85
| 563.54
| [[18/13]]
|-
| 86
| 570.17
|
|-
| 87
| 576.8
|
|-
| 88
| 583.43
| [[7/5]]
|-
| 89
| 590.06
| [[45/32]]
|-
| 90
| 596.69
| [[24/17]]
|-
| 91
| 603.31
| [[17/12]]
|-
| 92
| 609.94
| [[64/45]]
|-
| 93
| 616.57
| [[10/7]]
|-
| 94
| 623.2
|
|-
| 95
| 629.83
|
|-
| 96
| 636.46
| [[13/9]]
|-
| 97
| 643.09
|
|-
| 98
| 649.72
| [[16/11]]
|-
| 99
| 656.35
| [[19/13]]
|-
| 100
| 662.98
| [[22/15]]
|-
| 101
| 669.61
|
|-
| 102
| 676.24
| [[34/23]], [[65/44]]
|-
| 103
| 682.87
|
|-
| 104
| 689.5
|
|-
| 105
| 696.13
|
|-
| 106
| 702.76
| [[3/2]]
|-
| 107
| 709.39
|
|-
| 108
| 716.02
|
|-
| 109
| 722.65
|
|-
| 110
| 729.28
| [[32/21]]
|-
| 111
| 735.91
| [[26/17]]
|-
| 112
| 742.54
|
|-
| 113
| 749.17
|
|-
| 114
| 755.8
| [[65/42]]
|-
| 115
| 762.43
|
|-
| 116
| 769.06
|
|-
| 117
| 775.69
| [[36/23]]
|-
| 118
| 782.32
| [[11/7]]
|-
| 119
| 788.95
|
|-
| 120
| 795.58
| [[19/12]]
|-
| 121
| 802.21
| [[62/39]]
|-
| 122
| 808.84
|
|-
| 123
| 815.47
|
|-
| 124
| 822.1
| [[45/28]]
|-
| 125
| 828.73
|
|-
| 126
| 835.36
| [[34/21]]
|-
| 127
| 841.99
|
|-
| 128
| 848.62
| [[49/30]]
|-
| 129
| 855.25
|
|-
| 130
| 861.88
|
|-
| 131
| 868.51
| [[38/23]]
|-
| 132
| 875.14
| [[63/38]]
|-
| 133
| 881.77
|
|-
| 134
| 888.4
|
|-
| 135
| 895.03
| [[57/34]]
|-
| 136
| 901.66
| [[32/19]]
|-
| 137
| 908.29
|
|-
| 138
| 914.92
| [[39/23]], [[56/33]]
|-
| 139
| 921.55
| [[46/27]]
|-
| 140
| 928.18
| [[65/38]]
|-
| 141
| 934.81
|
|-
| 142
| 941.44
|
|-
| 143
| 948.07
|
|-
| 144
| 954.7
| [[33/19]]
|-
| 145
| 961.33
|
|-
| 146
| 967.96
| [[7/4]]
|-
| 147
| 974.59
|
|-
| 148
| 981.22
|
|-
| 149
| 987.85
| [[23/13]]
|-
| 150
| 994.48
|
|-
| 151
| 1001.1
|
|-
| 152
| 1007.73
| [[34/19]]
|-
| 153
| 1014.36
|
|-
| 154
| 1020.99
|
|-
| 155
| 1027.62
| [[38/21]]
|-
| 156
| 1034.25
| [[20/11]]
|-
| 157
| 1040.88
|
|-
| 158
| 1047.51
|
|-
| 159
| 1054.14
|
|-
| 160
| 1060.77
| [[24/13]]
|-
| 161
| 1067.4
| [[63/34]]
|-
| 162
| 1074.03
|
|-
| 163
| 1080.66
| [[28/15]]
|-
| 164
| 1087.29
| [[15/8]]
|-
| 165
| 1093.92
| [[32/17]]
|-
| 166
| 1100.55
| [[17/9]]
|-
| 167
| 1107.18
| [[36/19]]
|-
| 168
| 1113.81
| [[19/10]], [[40/21]]
|-
| 169
| 1120.44
| [[21/11]]
|-
| 170
| 1127.07
| [[23/12]]
|-
| 171
| 1133.7
| [[52/27]]
|-
| 172
| 1140.33
|
|-
| 173
| 1146.96
| [[64/33]]
|-
| 174
| 1153.59
|
|-
| 175
| 1160.22
|
|-
| 176
| 1166.85
| [[51/26]]
|-
| 177
| 1173.48
| [[63/32]], [[65/33]]
|-
| 178
| 1180.11
|
|-
| 179
| 1186.74
|
|-
| 180
| 1193.37
|
|-
| 181
| 1200
| [[2/1]]
|}
<nowiki/>*As a 23-limit temperament
 
== 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| 287 -181 }}
| {{Mapping| 181 287 }}
| −0.255
| 0.255
| 3.84
|-
| 2.3.5
| 2109375/2097152, {{monzo| 14 -22 9 }}
| {{Mapping| 181 287 420 }}
| +0.086
| 0.525
| 7.92
|-
| 2.3.5.7
| 2401/2400, 5120/5103, 390625/387072
| {{Mapping| 181 287 420 508 }}
| +0.142
| 0.465
| 7.01
|-
| 2.3.5.7.11
| 385/384, 1375/1372, 2200/2187, 4000/3993
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 181 287 420 508 626 670 740 769 }}
| +0.000
| 0.404
| 6.09
|}
 
=== 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
| 18\181
| 119.34
| 15/14
| [[Septidiasemi]]
|-
| 1
| 35\181
| 232.04
| 8/7
| [[Quadrawell]]
|-
| 1
| 39\181
| 258.56
| {{Monzo| -32 13 5 }}
| [[Lafa]]
|-
| 1
| 41\181
| 271.82
| 75/64
| [[Orson]]
|-
| 1
| 53\181
| 351.38
| 49/40
| [[Hemififths]] (7-limit)
|-
| 1
| 78\181
| 517.13
| 66/49
| [[Cutefourths]]
|-
| 1
| 88\181
| 583.43
| 7/5
| [[Cotritone]] (11-limit)
|}
<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
 
== Music ==
; [[Francium]]
* "Today Or Tomorrow?" from ''Questions'' (2024) – [https://open.spotify.com/track/24RikcnTP33a6v9vXyNPwh Spotify] | [https://francium223.bandcamp.com/track/today-or-tomorrow Bandcamp] | [https://www.youtube.com/watch?v=ipUyBAHIvlk YouTube] – slurpee in 181edo tuning
 
== See also ==
* [[181edo and stretched hemififths]]
 
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/181edo"