Hodge dual: Difference between revisions
Wikispaces>genewardsmith **Imported revision 289018701 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 289018761 - 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:genewardsmith|genewardsmith]] and made on <tt>2012-01-02 00: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-02 00:40:25 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>289018761</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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=Using the dual= | =Using the dual= | ||
The dual allows one to find the wedgie, which is a normalized multival, by wedging together monzos and then taking the dual. For instance from M = |0 3 -2 0>∧|-2 1 -1 1> | The dual allows one to find the wedgie, which is a normalized multival, by wedging together monzos and then taking the dual. For instance from M = |0 3 -2 0>∧|-2 1 -1 1>, which is ||6 -4 0 -1 3 -2>>, considered above, we may find the dual Mº as ||6 -4 0 -1 3 -2>>º = <<-2 -3 -1 0 4 6||. Normalizing this to a wedgie gives <<2 3 1 0 -4 -6||, the wedgie for bug temperament. Then if W is the wedgie for ennealimmal considered above, W∧Mº = <W|M> = 1. We can also take a multival, and use the dual to get a corresponding mulitmonzo, and then use the same method described on the [[abstract regular temperament]] page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.</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>The dual</title></head><body><!-- ws:start:WikiTextTocRule: | <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>The dual</title></head><body><!-- ws:start:WikiTextTocRule:8:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#The bracket">The bracket</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#The dual">The dual</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Computing the dual">Computing the dual</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Using the dual">Using the dual</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | ||
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Given a k-multival W, there is a <em>dual</em> k-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual k-multival Mº. The dual may be defined in terms of the bracket product relating multivals and multimonzos, which we discuss first.<br /> | Given a k-multival W, there is a <em>dual</em> k-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual k-multival Mº. The dual may be defined in terms of the bracket product relating multivals and multimonzos, which we discuss first.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Using the dual"></a><!-- ws:end:WikiTextHeadingRule:6 -->Using the dual</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Using the dual"></a><!-- ws:end:WikiTextHeadingRule:6 -->Using the dual</h1> | ||
The dual allows one to find the wedgie, which is a normalized multival, by wedging together monzos and then taking the dual. For instance from M = |0 3 -2 0&gt;∧|-2 1 -1 1&gt; | The dual allows one to find the wedgie, which is a normalized multival, by wedging together monzos and then taking the dual. For instance from M = |0 3 -2 0&gt;∧|-2 1 -1 1&gt;, which is ||6 -4 0 -1 3 -2&gt;&gt;, considered above, we may find the dual Mº as ||6 -4 0 -1 3 -2&gt;&gt;º = &lt;&lt;-2 -3 -1 0 4 6||. Normalizing this to a wedgie gives &lt;&lt;2 3 1 0 -4 -6||, the wedgie for bug temperament. Then if W is the wedgie for ennealimmal considered above, W∧Mº = &lt;W|M&gt; = 1. We can also take a multival, and use the dual to get a corresponding mulitmonzo, and then use the same method described on the <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.</body></html></pre></div> | ||