Hodge dual: Difference between revisions
Wikispaces>clumma **Imported revision 535153084 - Original comment: ** |
Wikispaces>clumma **Imported revision 535153114 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:clumma|clumma]] and made on <tt>2014-12-14 22:40: | : This revision was by author [[User:clumma|clumma]] and made on <tt>2014-12-14 22:40:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>535153114</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> | ||
<h4>Original Wikitext content:</h4> | <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"> | <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">Given a k-multival W, there is a //dual// (n-k)-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual (n-k)-multival Mº. The dual may be defined in terms of the bracket product relating multivals and multimonzos, which we discuss first. | ||
Given a k-multival W, there is a //dual// (n-k)-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual (n-k)-multival Mº. The dual may be defined in terms of the bracket product relating multivals and multimonzos, which we discuss first. | |||
=The bracket= | =The bracket= | ||
| Line 22: | Line 20: | ||
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> | 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> | <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>Given a k-multival W, there is a <em>dual</em> (n-k)-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual (n-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> (n-k)-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual (n-k)-multival Mº. The dual may be defined in terms of the bracket product relating multivals and multimonzos, which we discuss first.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="The bracket"></a><!-- ws:end:WikiTextHeadingRule:0 -->The bracket</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="The bracket"></a><!-- ws:end:WikiTextHeadingRule:0 -->The bracket</h1> | ||