255edo: Difference between revisions

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-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 3 × 5 × 17
+{{ED intro}}
-| Step size = 4.70589¢
-| Fifth = 149\255 (701.18¢)
-| Semitones = 23:20 (108.24¢ : 94.12¢)
-| Consistency = 11
-}}
-{{EDO intro|255}}
 == Theory ==
-et 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]].
+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}}
+=== 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" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal 8ve <br>stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 26: / Line 24: @@
 |-
 | 2.3
-| {{monzo| -404 255 }}
+| {{Monzo| -404 255 }}
-| [{{val| 255 404 }}]
+| {{Mapping| 255 404 }}
 | +0.246
 | 0.246
@@ Line 33: / Line 31: @@
 |-
 | 2.3.5
-| {{monzo| 8 14 -13 }}, {{monzo| -36 11 8 }}
+| {{Monzo| 8 14 -13 }}, {{monzo| -36 11 8 }}
-| [{{val| 255 404 592 }}]
+| {{Mapping| 255 404 592 }}
 | +0.226
 | 0.203
@@ Line 40: / Line 38: @@
 |-
 | 2.3.5.7
-| 1687/16807, 19683/19600, 65625/65536
+| 16875/16807, 19683/19600, 65625/65536
-| [{{val| 255 404 592 716 }}]
+| {{Mapping| 255 404 592 716 }}
 | +0.117
 | 0.257
@@ Line 48: / Line 46: @@
 | 2.3.5.7.11
 | 540/539, 1375/1372, 8019/8000, 65625/65536
-| [{{val| 255 404 592 716 882 }}]
+| {{Mapping| 255 404 592 716 882 }}
 | +0.136
 | 0.233
@@ Line 56: / Line 54: @@
 === Rank-2 temperaments ===
 {| 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 octave
+|-
-! Generator<br>(reduced)
+! Periods<br>per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 92: / Line 91: @@
 | 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
@@ Line 97: / Line 102: @@
 | 498.82<br>(18.82)
 | 4/3<br>(81/80)
-| [[Pental]] (5-limit)
+| [[Quintile]]
 |-
 | 17
 | 53\255<br>(7\255)
 | 249.41<br>(32.94)
-| {{monzo| -25 -9 17 }}<br>(1990656/1953125)
+| {{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:Equal divisions of the octave]]
 [[Category:Mirkat]]