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The '''183 equal divisions of the octave''' ('''183edo'''), or the '''183(-tone) equal temperament''' ('''183tet''', '''183et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 183 [[equal]] parts of 6.557 [[cent]]s each.  
{{Infobox ET}}
The '''183 equal divisions of the octave''' ('''183edo'''), or the '''183(-tone) equal temperament''' ('''183tet''', '''183et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 183 [[equal]] parts of about 6.56 [[cent]]s each, a size close to [[243/242]], the rastma.  


== Theory ==
== Theory ==
183edo is notable as a higher limit system, especially when 7 is left out of the picture. It tempers out the [[schisma]], 32805/32768, in the [[5-limit]]. In the [[7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit]], it tempers out [[540/539]], [[3025/3024]] and [[8019/8000]]; in the [[13-limit]], [[351/350]] and [[676/675]]; in the [[17-limit]] 442/441, 561/560 and 715/714; and in the [[19-limit]] 456/455. It is the [[optimal patent val]] for 13-, 17- and 19-limit [[mirkat]] temperament, the 72&183 temperament, and an excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]].
183edo is notable as a higher-limit system, [[consistency|distinctly consistent]] in the [[17-odd-limit]], or the no-19 no-31 [[33-odd-limit]]. It has especially low errors in ''all'' [[prime limit]]s from 11 to 29, although its bad rendering of [[19/1|19]] makes it fail to be consistent in the [[19-odd-limit]]. It is however a strong no-19's [[29-limit]] system with the addition of an essentially perfectly accurate prime [[43/1|43]].  


As a no-sevens temperament, it tempers out [[32805/32768]], 5632/5625, [[8019/8000]], [[676/675]], 4425/4424, 6656/6655, [[936/935]], [[1089/1088]], and 1377/1375.
As an equal temperament, 183et [[tempering out|tempers out]] the [[schisma]] in the [[5-limit]]. In the [[7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit]], it tempers out [[540/539]], 1375/1372, [[3025/3024]], [[5632/5625]], and [[8019/8000]]; in the [[13-limit]], [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[1573/1568]], [[2080/2079]], [[4096/4095]], [[4225/4224]], and [[6656/6655]]; in the [[17-limit]] [[442/441]], [[561/560]], [[715/714]], [[936/935]], [[1089/1088]], and [[1156/1155]]; and in the [[19-limit]] [[456/455]]. It is the [[optimal patent val]] for 13- and 17-limit [[mirkat]], the {{nowrap|72 & 111}} temperament, and an excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]]. It allows [[essentially tempered chord]] including [[ratwolfsmic chords]], [[swetismic chords]], [[squbemic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit.
 
It is even stronger if 7 is left out of the picture. As a no-7 temperament, it tempers out 5632/5625, 8019/8000, 676/675, 4225/4224, 6656/6655, 936/935, 1089/1088, and 1377/1375.


=== Prime harmonics ===
=== Prime harmonics ===
183edo is notable as having especially low error in ''all'' [[prime limit]]s from 11 to 29 for EDOs in the 100 to 200 range, compared using a variety of metrics (prime error punishments), although it has a bad 19 which causes it to fail to be consistent in the [[19-odd-limit]]. It is however a strong no-19's system, being consistent in the no-19's no-35's 29-prime-limited 45-odd-limit add-43. (The prime 43 is added in the set of odd harmonics due to its essentially perfect accuracy. The harmonic 35 is excluded due to the sharpness of 7 compounding and causing inconsistency in ''some'' cases such as for 39/35.) It can also be considered to model the 2.17.29.43 subgroup with extreme accuracy.
{{Harmonics in equal|183|columns=11}}
{{Harmonics in equal|183|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 183edo (continued)}}
 
=== Subsets and supersets ===
Since 183 factors into primes as {{nowrap| 3 × 61 }}, 183edo contains [[3edo]] and [[61edo]] as its subsets.
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
 
== 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| -290 183 }}
| {{mapping| 183 290 }}
| +0.0996
| 0.100
| 1.52
|-
| 2.3.5
| 32805/32768, {{val| 10 23 -20 }}
| {{mapping| 183 290 425 }}
| −0.0157
| 0.182
| 2.78
|-
| 2.3.5.7
| 6144/6125, 16875/16807, 19683/19600
| {{mapping| 183 290 425 514 }}
| −0.1601
| 0.296
| 4.51
|-
| 2.3.5.7.11
| 540/539, 1375/1372, 5632/5625, 8019/8000
| {{mapping| 183 290 425 514 633 }}
| −0.0993
| 0.291
| 4.44
|-
| 2.3.5.7.11.13
| 351/350, 540/539, 676/675, 1375/1372, 4096/4095
| {{mapping| 183 290 425 514 633 677 }}
| −0.0295
| 0.308
| 4.70
|-
| 2.3.5.7.11.13.17
| 351/350, 442/441, 540/539, 561/560, 1375/1372, 4096/4095
| {{mapping| 183 290 425 514 633 677 748 }}
| −0.0240
| 0.286
| 4.36
|}
* 183et has lower absolute errors in the 13-, 17-, 19-, and 23-limit than any previous equal temperaments, after [[130edo|130]], [[171edo|171]], [[161edo|161]], and [[159edo|159]], respectively. In the 13-, 19-, and 23-limit it is superseded by [[190edo|190g]]. In the 17-limit, where it is the strongest, by [[217edo|217]].
 
=== 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
| 10\183
| 65.57
| 27/26
| [[Luminal]]
|-
| 1
| 16\183
| 104.92
| 17/16
| [[Septendesemi]]
|-
| 1
| 17\183
| 111.48
| 16/15
| [[Stockhausenic]]
|-
| 1
| 38\183
| 249.18
| 15/13
| [[Hemischis]]
|-
| 1
| 58\183
| 380.33
| 56/45
| [[Quanharuk]]
|-
| 1
| 59\183
| 386.89
| 5/4
| [[Grendel]]
|-
| 1
| 76\183
| 498.36
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|-
| 1
| 77\183
| 504.92
| 104976/78125
| [[Countermeantone]]
|-
| 3
| 21\183
| 137.70
| 13/12
| [[Avicenna]]
|-
| 3
| 24\183
| 157.38
| 35/32
| [[Nessafof]]
|-
| 3
| 28\183
| 183.61
| 10/9
| [[Mirkat]]
|-
| 3
| 38\183<br>(23\183)
| 249.18<br>(150.82)
| 15/13<br>(12/11)
| [[Hemiterm]]
|-
| 3
| 76\183<br>(15\183)
| 498.36<br>(98.36)
| 4/3<br>(200/189)
| [[Term]] / terminator
|-
| 61
| 38\183<br>(2\183)
| 249.18<br>(13.11)
| 13750/11907<br>(?)
| [[Promethium]]
|}
<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


{{Primes in edo|183|columns=10}}
== Music ==
; [[birdshite stalactite]]
* from ''meticulous clutter'' (2022)
** "cursed owl windchimes" – [https://open.spotify.com/track/0A9z6gw5HuNWS7eYjvtCa1 Spotify] | [https://birdshitestalactite.bandcamp.com/track/cursed-owl-windchimes Bandcamp] | [https://www.youtube.com/watch?v=vyTbwHhdoXE YouTube] – madagascar[19] in 183edo tuning
** "octopus bones" – [https://open.spotify.com/track/6TUm7uXjVaO6MGACWswoL7 Spotify] | [https://birdshitestalactite.bandcamp.com/track/octopus-bones Bandcamp] | [https://www.youtube.com/watch?v=ej80dEpC-DQ YouTube] – madagascar[19] in 183edo tuning and parakleismic[19] in 61edo tuning
* "lemon drizzle" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/3dBXs1KjzTjSWJ7KF4BjlK Spotify] | [https://birdshitestalactite.bandcamp.com/track/lemon-drizzle Bandcamp] | [https://www.youtube.com/watch?v=VJ_bAvWB8W0 YouTube]


[[Category:Equal divisions of the octave]]
[[Category:Borneo]]
[[Category:Borneo]]
[[Category:Madagascar]]
[[Category:Madagascar]]
[[Category:Mirkat]]
[[Category:Mirkat]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/183edo"