Valinorsmic clan: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
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 23:01:12 UTC</tt>.<br>
: The original revision id was <tt>268616224</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">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.)


=Rank four valinorsmic temperament=
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]].
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|]
Discussed elsewhere are:  
EDOs: 22, 31, 46, 53, 58, 80, 111
* [[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]]


=Manwe=
Considered below is manwe, besides the no-3 subgroup temperament. For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (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|]
== No-3 valinorsmic ==
EDOs: 15, 31, 46, 77, 80, 111
[[Subgroup]]: 2.5.7.11
Badness: 0.000679


==13-limit==
[[Comma list]]: 176/175
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|]
{{Mapping|legend=2| 1 0 0 -4 | 0 1 0 2 | 0 0 1 1 }}
EDOs: 31, 46, 77, 80, 111
Badness: 0.001147


=Varda=
: sval mapping generators: ~2, ~5, ~7
Commas: 176/175, 896/891


Badness: 0.000728
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 389.7641, ~7/4 = 972.1113


=Mandos=
{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}
Commas: 176/175, 243/242


Badness: 0.000781
[[Badness]]: 1.73 × 10<sup>-6</sup>


==13-limit==
== Manwe ==
Commas: 144/143, 176/175, 243/242
[[Subgroup]]: 2.3.5.7.11


Badness: 0.000923
[[Comma list]]: 176/175, 1331/1323


{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}


</pre></div>
: mapping generators: ~2, ~3, ~5
<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;Valinorismic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&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;
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }}
&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;
[[Badness]]: 0.679 × 10<sup>-3</sup>
Comma: 176/175&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
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;
Subgroup: 2.3.5.7.11.13
EDOs: 22, 31, 46, 53, 58, 80, 111&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 176/175, 1331/1323, 351/350
&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;
 
Commas: 176/175, 1331/1323&lt;br /&gt;
Mapping: {{mapping| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}
&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;
{{Optimal ET sequence|legend=1| 31, 46, 77, 80, 111 }}
EDOs: 15, 31, 46, 77, 80, 111&lt;br /&gt;
 
Badness: 0.000679&lt;br /&gt;
Badness: 1.147 × 10<sup>-3</sup>
&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;
[[Category:Temperament clans]]
Commas: 176/175, 1331/1323, 351/350&lt;br /&gt;
[[Category:Pages with mostly numerical content]]
&lt;br /&gt;
[[Category:Valinorsmic clan| ]] <!-- main article -->
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;
[[Category:Valinorsmic| ]] <!-- key article -->
EDOs: 31, 46, 77, 80, 111&lt;br /&gt;
[[Category:Rank 3]]
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;
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;
Badness: 0.000781&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;
Badness: 0.000923&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 00:39, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The valinorsmic clan of rank-3 temperaments 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).

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.

Discussed elsewhere are:

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

No-3 valinorsmic

Subgroup: 2.5.7.11

Comma list: 176/175

Sval mapping: [⟨1 0 0 -4], ⟨0 1 0 2], ⟨0 0 1 1]]

sval mapping generators: ~2, ~5, ~7

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 389.7641, ~7/4 = 972.1113

Optimal ET sequence: 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd

Badness: 1.73 × 10-6

Manwe

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1331/1323

Mapping: [⟨1 0 0 12 8], ⟨0 1 0 3 3], ⟨0 0 1 -6 -4]]

mapping generators: ~2, ~3, ~5

Optimal ET sequence: 15, 31, 46, 77, 80, 111

Badness: 0.679 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 1331/1323, 351/350

Mapping: [⟨1 0 0 12 8], ⟨0 1 0 3 3], ⟨0 0 1 -6 -4]]

Optimal ET sequence: 31, 46, 77, 80, 111

Badness: 1.147 × 10-3

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