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Valinorsmic clan: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''valinorsmic clan''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the valinorsma, [[176/175]], which equates 48/35 with 15/11 (rather than 11/8, which is what the keenanisma [[385/384]] does).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-25 23:23:31 UTC</tt>.<br>
: The original revision id was <tt>268621406</tt>.<br>
: The revision comment was: <tt></tt><br>
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>
<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">[[toc|flat]]


The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds [[121/120]] as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. Julius, a.k.a. varda, adds [[896/891]], slicing the first generator in two with a semi-octave period. Nickel adds [[36/35]] as well as [[45/44]]. Ares adds [[64/63]]. Minerva adds [[99/98]]. Thrush adds [[126/125]]. These slice the last generator in two. Guanyin adds [[540/539]], slicing the first generator in three. Manwe adds [[1331/1323]], slicing the last generator in three. Clio adds [[81/80]], slicing the first generator in four. Lono adds [[5120/5103]], slicing the last generator in six. Mandos adds [[243/242]]. Shrusus adds [[245/243]]. These slice the last generator in five. Ulmo adds [[2200/2187]], slicing the last generator in seven. Finally, draco adds [[19683/19600]], slicing the last generator in nine. Most of these have neat extensions to the 13-limit via tempering out both [[351/350]] and [[352/351]].


The valinorsmic rank three temperaments temper out the valinorsma, 176/175, which equates 48/35 with 15/11 (rather than 11/8, which is what the keenanisma 385/384 does.)
Discussed elsewhere are:
* [[Zeus]] (+121/120) → [[Porwell family #Zeus|Porwell family]] and [[Biyatismic clan #Zeus|Biyatismic clan]]
* ''[[Julius]]'' or ''[[varda]]'' (+896/891) → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]
* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]
* [[Ares]] (+64/63 or 100/99) → [[Archytas family #Ares|Archytas family]]
* ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]
* [[Thrush]] (+126/125) → [[Starling family #Thrush|Starling family]]
* ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]
* ''[[Clio]]'' (+81/80) → [[Didymus rank three family #Clio|Didymus rank-3 family]]
* ''[[Lono]]'' (+5120/5103) → [[Hemifamity family #Lono|Hemifamity family]]
* ''[[Mandos]]'' (+243/242) → [[Rastmic rank three clan #Mandos|Rastmic rank-3 clan]]
* ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]
* ''[[Ulmo]]'' (+2200/2187) → [[Ragismic family #Ulmo|Ragismic family]]
* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]


=Rank four valinorsmic temperament=
Considered below is manwe, besides the no-3 subgroup temperament. For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].
Comma: 176/175


Map: [&lt;1 0 0 0 -4|, &lt;0 1 0 0 0|, &lt;0 0 1 0 2|, &lt;0 0 0 1 1|]
== No-3 valinorsmic ==
EDOs: 22, 31, 46, 53, 58, 80, 111
[[Subgroup]]: 2.5.7.11


=Manwe=
[[Comma list]]: 176/175
Commas: 176/175, 1331/1323


Map: [&lt;1 0 0 12 8|, &lt;0 1 0 3 3|, &lt;0 0 1 -6 -4|]
{{Mapping|legend=2| 1 0 0 -4 | 0 1 0 2 | 0 0 1 1 }}
EDOs: 15, 31, 46, 77, 80, 111
Badness: 0.000679


==13-limit==
: sval mapping generators: ~2, ~5, ~7
Commas: 176/175, 1331/1323, 351/350


Map: [&lt;1 0 0 12 8|, &lt;0 1 0 3 3|, &lt;0 0 1 -6 -4|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 389.7641, ~7/4 = 972.1113
EDOs: 31, 46, 77, 80, 111
Badness: 0.001147


=Varda=
{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}
Commas: 176/175, 896/891


Map: [&lt;2 0 11 0 14|, &lt;0 1 -2 0 -4|, &lt;0 0 0 1 1|]
[[Badness]]: 1.73 × 10<sup>-6</sup>
EDOs: 12, 22, 46, 58, 80, 126
Badness: 0.000728


=Mandos=
== Manwe ==
Commas: 176/175, 243/242
[[Subgroup]]: 2.3.5.7.11


Map: [&lt;1 1 0 6 2|, &lt;0 2 0 5 5|, &lt;0 0 1 -2 0|]
[[Comma list]]: 176/175, 1331/1323
EDOs: 7, 24, 31, 58, 89
Badness: 0.000751


==13-limit==
{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}
Commas: 144/143, 176/175, 243/242


Map: [&lt;1 1 0 6 2 4|, &lt;0 2 0 5 5 -1|, &lt;0 0 1 -2 0 0|]
: mapping generators: ~2, ~3, ~5
EDOs: 7, 24, 31, 58
Badness: 0.000923


{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }}


[[Badness]]: 0.679 × 10<sup>-3</sup>


</pre></div>
=== 13-limit ===
<h4>Original HTML content:</h4>
Subgroup: 2.3.5.7.11.13
<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;Valinorismic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Rank four valinorsmic temperament"&gt;Rank four valinorsmic temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Manwe"&gt;Manwe&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Varda"&gt;Varda&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Mandos"&gt;Mandos&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
Comma list: 176/175, 1331/1323, 351/350
&lt;br /&gt;
 
The valinorsmic rank three temperaments temper out the valinorsma, 176/175, which equates 48/35 with 15/11 (rather than 11/8, which is what the keenanisma 385/384 does.)&lt;br /&gt;
Mapping: {{mapping| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Rank four valinorsmic temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Rank four valinorsmic temperament&lt;/h1&gt;
{{Optimal ET sequence|legend=1| 31, 46, 77, 80, 111 }}
Comma: 176/175&lt;br /&gt;
 
&lt;br /&gt;
Badness: 1.147 × 10<sup>-3</sup>
Map: [&amp;lt;1 0 0 0 -4|, &amp;lt;0 1 0 0 0|, &amp;lt;0 0 1 0 2|, &amp;lt;0 0 0 1 1|]&lt;br /&gt;
 
EDOs: 22, 31, 46, 53, 58, 80, 111&lt;br /&gt;
[[Category:Temperament clans]]
&lt;br /&gt;
[[Category:Pages with mostly numerical content]]
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Manwe"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Manwe&lt;/h1&gt;
[[Category:Valinorsmic clan| ]] <!-- main article -->
Commas: 176/175, 1331/1323&lt;br /&gt;
[[Category:Valinorsmic| ]] <!-- key article -->
&lt;br /&gt;
[[Category:Rank 3]]
Map: [&amp;lt;1 0 0 12 8|, &amp;lt;0 1 0 3 3|, &amp;lt;0 0 1 -6 -4|]&lt;br /&gt;
EDOs: 15, 31, 46, 77, 80, 111&lt;br /&gt;
Badness: 0.000679&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Manwe-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
Commas: 176/175, 1331/1323, 351/350&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 12 8|, &amp;lt;0 1 0 3 3|, &amp;lt;0 0 1 -6 -4|]&lt;br /&gt;
EDOs: 31, 46, 77, 80, 111&lt;br /&gt;
Badness: 0.001147&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Varda"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Varda&lt;/h1&gt;
Commas: 176/175, 896/891&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 0 11 0 14|, &amp;lt;0 1 -2 0 -4|, &amp;lt;0 0 0 1 1|]&lt;br /&gt;
EDOs: 12, 22, 46, 58, 80, 126&lt;br /&gt;
Badness: 0.000728&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Mandos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Mandos&lt;/h1&gt;
Commas: 176/175, 243/242&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 1 0 6 2|, &amp;lt;0 2 0 5 5|, &amp;lt;0 0 1 -2 0|]&lt;br /&gt;
EDOs: 7, 24, 31, 58, 89&lt;br /&gt;
Badness: 0.000751&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Mandos-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
Commas: 144/143, 176/175, 243/242&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 1 0 6 2 4|, &amp;lt;0 2 0 5 5 -1|, &amp;lt;0 0 1 -2 0 0|]&lt;br /&gt;
EDOs: 7, 24, 31, 58&lt;br /&gt;
Badness: 0.000923&lt;/body&gt;&lt;/html&gt;</pre></div>
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