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255edo: Difference between revisions

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**Imported revision 240468039 - Original comment: **
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-08 00:00:49 UTC</tt>.<br>
 
: The original revision id was <tt>240468039</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
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]].
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
=== Prime harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //255 equal division// divides the octave into 255 equal parts of 4.706 cents each. It tempers out the parakleisma, |8 14 -13&gt;, and the septendecima, |-52 -17 34&gt;, in the 5-limit. In the 7-limit it tempers out cataharry, 19683/19600, mirkwai, 16875/16807 and horwell, 65625/65536, so that it supports [[Mirkwai clan#Mirkat|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.</pre></div>
{{Harmonics in equal|255}}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;255edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;255 equal division&lt;/em&gt; divides the octave into 255 equal parts of 4.706 cents each. It tempers out the parakleisma, |8 14 -13&amp;gt;, and the septendecima, |-52 -17 34&amp;gt;, in the 5-limit. In the 7-limit it tempers out cataharry, 19683/19600, mirkwai, 16875/16807 and horwell, 65625/65536, so that it supports &lt;a class="wiki_link" href="/Mirkwai%20clan#Mirkat"&gt;mirkat temperament&lt;/a&gt;, and in fact provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;. 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.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== 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]]
Retrieved from "https://en.xen.wiki/w/255edo"