User:AthiTrydhen/15-limit tonality diamond: Difference between revisions
Wikispaces>Praimhin **Imported revision 588772064 - Original comment: ** |
Wikispaces>Praimhin **Imported revision 588772130 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:Praimhin|Praimhin]] and made on <tt>2016-08-04 01: | : This revision was by author [[User:Praimhin|Praimhin]] and made on <tt>2016-08-04 01:23:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>588772130</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators: | The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators: | ||
* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9 | * Transformation //R//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9 | ||
* 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8 | * Transformation //S//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8 | ||
* 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8 | * Transformation //S'//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8 | ||
* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13 | * Transformation //T//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13 | ||
These generators have the relations //R//² = //S//² = //T//² = //S//'² = I, (//SS//')³ = I, //RS// = //SR//, //RS//' = //S//'//R//, and //T// commutes with the three other generators. Thus the symmetry group is isomorphic to //S//₃ × //C//₂². | |||
===Orbits and Invariant Subsets=== | ===Orbits and Invariant Subsets=== | ||
[ | The [[Hendrix diamond]] is invariant under action by //R,// //S//' and //T//, and the images of the action of //S// and //S//² on the Hendrix diamond are the [[11-Hendrix diamond]] and [[13-Hendrix diamond]] respectively. | ||
Two other interesting invariant subsets are the 5-limit tonality diamond and the tonality diamond constructed from the harmonics 1, 3, 5, 9 and 15.</pre></div> | |||
<h4>Original HTML content:</h4> | <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"><html><head><title>15-limit tonality diamond</title></head><body>The <strong>15-limit tonality diamond</strong> has the following notes:<br /> | <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"><html><head><title>15-limit tonality diamond</title></head><body>The <strong>15-limit tonality diamond</strong> has the following notes:<br /> | ||
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The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:<br /> | The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:<br /> | ||
<ul><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul><br /> | <ul><li>Transformation <em>R</em>: 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>Transformation <em>S</em>: 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>Transformation <em>S'</em>: 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>Transformation <em>T</em>: 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul><br /> | ||
These generators have the relations <em>R</em>² = <em>S</em>² = <em>T</em>² = <em>S</em>'² = I, (<em>SS</em>')³ = I, <em>RS</em> = <em>SR</em>, <em>RS</em>' = <em>S</em>'<em>R</em>, and <em>T</em> commutes with the three other generators. Thus the symmetry group is isomorphic to <em>S</em>₃ × <em>C</em>₂².<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Symmetry group-Orbits and Invariant Subsets"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orbits and Invariant Subsets</h3> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Symmetry group-Orbits and Invariant Subsets"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orbits and Invariant Subsets</h3> | ||
The <a class="wiki_link" href="/Hendrix%20diamond">Hendrix diamond</a> is invariant under action by <em>R,</em> <em>S</em>' and <em>T</em>, and the images of the action of <em>S</em> and <em>S</em>² on the Hendrix diamond are the <a class="wiki_link" href="/11-Hendrix%20diamond">11-Hendrix diamond</a> and <a class="wiki_link" href="/13-Hendrix%20diamond">13-Hendrix diamond</a> respectively.<br /> | |||
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Two other interesting invariant subsets are the 5-limit tonality diamond and the tonality diamond constructed from the harmonics 1, 3, 5, 9 and 15.</body></html></pre></div> | |||