183edo: Difference between revisions
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== Theory == | == Theory == | ||
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 [[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. | 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 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. | 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 === | ||
{{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 == | == Regular temperament properties == | ||
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! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 75: | Line 80: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 86: | Line 91: | ||
| 27/26 | | 27/26 | ||
| [[Luminal]] | | [[Luminal]] | ||
|- | |||
| 1 | |||
| 16\183 | |||
| 104.92 | |||
| 17/16 | |||
| [[Septendesemi]] | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 115: | Line 126: | ||
| 498.36 | | 498.36 | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] | | [[Helmholtz (temperament)|Helmholtz]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 142: | Line 153: | ||
|- | |- | ||
| 3 | | 3 | ||
| 38\183<br | | 38\183<br>(23\183) | ||
| 249.18<br | | 249.18<br>(150.82) | ||
| 15/13<br | | 15/13<br>(12/11) | ||
| [[Hemiterm]] | | [[Hemiterm]] | ||
|- | |- | ||
| 3 | | 3 | ||
| 76\183<br | | 76\183<br>(15\183) | ||
| 498.36<br | | 498.36<br>(98.36) | ||
| 4/3<br | | 4/3<br>(200/189) | ||
| [[Term]] / terminator | | [[Term]] / terminator | ||
|- | |- | ||
| 61 | | 61 | ||
| 38\183<br | | 38\183<br>(2\183) | ||
| 249.18<br | | 249.18<br>(13.11) | ||
| 13750/11907<br | | 13750/11907<br>(?) | ||
| [[Promethium]] | | [[Promethium]] | ||
|} | |} | ||
<nowiki />* [[Normal | <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 == | == Music == | ||