255edo: Difference between revisions
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{{Infobox ET | {{Infobox ET}} | ||
{{ED intro}} | |||
}} | |||
{{ | |||
== Theory == | == Theory == | ||
The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[septendecima]], {{monzo| -52 -17 34 }}, in the 5-limit. In the 7-limit it tempers out [[cataharry]], 19683/19600, [[mirkwai]], 16875/16807 and [[horwell]], 65625/65536, so that it [[support]]s the [[mirkat]] temperament, and in fact provides the [[optimal patent val]]. It also gives the optimal patent val for mirkat in the 11-limit, tempering out [[540/539]], 1375/1372, [[3025/3024]] and [[8019/8000]]. In the 13-limit it tempers out [[847/845]], [[625/624]], [[1575/1573]] and [[1716/1715]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|255}} | {{Harmonics in equal|255}} | ||
=== Subsets and supersets === | |||
Since 255 factors into {{factorization|255}}, 255edo has subset edos {{EDOs| 3, 5, 15, 17, 51, and 85 }}. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | |- | ||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 26: | Line 24: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| -404 255 }} | ||
| | | {{Mapping| 255 404 }} | ||
| +0.246 | | +0.246 | ||
| 0.246 | | 0.246 | ||
| Line 33: | Line 31: | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| 8 14 -13 }}, {{monzo| -36 11 8 }} | ||
| | | {{Mapping| 255 404 592 }} | ||
| +0.226 | | +0.226 | ||
| 0.203 | | 0.203 | ||
| Line 40: | Line 38: | ||
|- | |- | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| | | 16875/16807, 19683/19600, 65625/65536 | ||
| | | {{Mapping| 255 404 592 716 }} | ||
| +0.117 | | +0.117 | ||
| 0.257 | | 0.257 | ||
| Line 48: | Line 46: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 540/539, 1375/1372, 8019/8000, 65625/65536 | | 540/539, 1375/1372, 8019/8000, 65625/65536 | ||
| | | {{Mapping| 255 404 592 716 882 }} | ||
| +0.136 | | +0.136 | ||
| 0.233 | | 0.233 | ||
| Line 56: | Line 54: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per | |- | ||
! Generator | ! Periods<br>per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br>ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 97: | Line 96: | ||
| 249.41<br>(9.41) | | 249.41<br>(9.41) | ||
| 81/70<br>(176/175) | | 81/70<br>(176/175) | ||
| [[ | | [[Hemiquintile]] / hemiquint (255) / hemiquintilis (255f) | ||
|- | |- | ||
| 5 | | 5 | ||
| Line 103: | Line 102: | ||
| 498.82<br>(18.82) | | 498.82<br>(18.82) | ||
| 4/3<br>(81/80) | | 4/3<br>(81/80) | ||
| [[ | | [[Quintile]] | ||
|- | |- | ||
| 17 | | 17 | ||
| 53\255<br>(7\255) | | 53\255<br>(7\255) | ||
| 249.41<br>(32.94) | | 249.41<br>(32.94) | ||
| {{ | | {{Monzo| -25 -9 17 }}<br>(1990656/1953125) | ||
| [[Chlorine]] (5-limit) | | [[Chlorine]] (5-limit) | ||
|} | |} | ||
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
[[Category:Mirkat]] | [[Category:Mirkat]] | ||