255edo: Difference between revisions

← Older edit

VisualWikitext

@@ Line 1: / Line 1: @@
-The '''255 equal division''' divides the octave into 255 equal parts of 4.706 cents each. It 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]].
+{{Infobox ET}}
+{{ED intro}}
+== 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 ===
 {{Harmonics in equal|255}}
-[[Category:Equal divisions of the octave]]
+=== Subsets and supersets ===
+Since 255 factors into {{factorization|255}}, 255edo has subset edos {{EDOs| 3, 5, 15, 17, 51, and 85 }}.
+== 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| -404 255 }}
+| {{Mapping| 255 404 }}
+| +0.246
+| 0.246
+| 5.22
+|-
+| 2.3.5
+| {{Monzo| 8 14 -13 }}, {{monzo| -36 11 8 }}
+| {{Mapping| 255 404 592 }}
+| +0.226
+| 0.203
+| 4.30
+|-
+| 2.3.5.7
+| 16875/16807, 19683/19600, 65625/65536
+| {{Mapping| 255 404 592 716 }}
+| +0.117
+| 0.257
+| 5.46
+|-
+| 2.3.5.7.11
+| 540/539, 1375/1372, 8019/8000, 65625/65536
+| {{Mapping| 255 404 592 716 882 }}
+| +0.136
+| 0.233
+| 4.95
+|}
+=== 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
+| 39\255
+| 183.53
+| 10/9
+| [[Mirkat]] (255f)
+|-
+| 1
+| 52\255
+| 244.71
+| 15/13
+| [[Subsemifourth]] (255)
+|-
+| 1
+| 67\255
+| 315.29
+| 6/5
+| [[Parakleismic]] (5-limit)
+|-
+| 1
+| 74\255
+| 348.24
+| 11/9
+| [[Eris]] (255)
+|-
+| 3
+| 82\255<br>(3\255)
+| 385.88<br>(14.12)
+| 5/4<br>(126/125)
+| [[Mutt]] (7-limit)
+|-
+| 5
+| 53\255<br>(2\255)
+| 249.41<br>(9.41)
+| 81/70<br>(176/175)
+| [[Hemiquintile]] / hemiquint (255) / hemiquintilis (255f)
+|-
+| 5
+| 106\255<br>(4\255)
+| 498.82<br>(18.82)
+| 4/3<br>(81/80)
+| [[Quintile]]
+|-
+| 17
+| 53\255<br>(7\255)
+| 249.41<br>(32.94)
+| {{Monzo| -25 -9 17 }}<br>(1990656/1953125)
+| [[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]]