Hodge dual: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2016-05-18 14:55:25 UTC</tt>.<br>

: This revision was by author [[User:clumma|clumma]] and made on <tt>2016-05-18 15:19:24 UTC</tt>.<br>

: The original revision id was <tt>~~583495891~~</tt>.<br>

: The original revision id was <tt>583498319</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">Given n basis elements (i.e. the number of primes in a prime limit) and a k-multival W in this basis, there is a //dual// (n-k)-multimonzo Wº ~~(where º denotes the complement)~~. 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.

<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 n basis elements (i.e. the number of primes in a prime limit) and a k-multival W in this basis, 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=

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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>

<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 n basis elements (i.e. the number of primes in a prime limit) and a k-multival W in this basis, there is a <em>dual</em> (n-k)-multimonzo Wº ~~(where º denotes the complement)~~. 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 />

<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 n basis elements (i.e. the number of primes in a prime limit) and a k-multival W in this basis, 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 />

<h1 id="toc0"><a name="The bracket"></a>The bracket</h1>

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2016-05-18 14:55:25 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2016-05-18 15:19:24 UTC</tt>.<br>
-: The original revision id was <tt>583495891</tt>.<br>
+: The original revision id was <tt>583498319</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">Given n basis elements (i.e. the number of primes in a prime limit) and a k-multival W in this basis, there is a //dual// (n-k)-multimonzo Wº (where º denotes the complement). 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.
+<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 n basis elements (i.e. the number of primes in a prime limit) and a k-multival W in this basis, 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=
@@ Line 20: / 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&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 [[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>
-<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;The dual&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Given n basis elements (i.e. the number of primes in a prime limit) and a k-multival W in this basis, there is a &lt;em&gt;dual&lt;/em&gt; (n-k)-multimonzo Wº (where º denotes the complement). 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.&lt;br /&gt;
+<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;The dual&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Given n basis elements (i.e. the number of primes in a prime limit) and a k-multival W in this basis, there is a &lt;em&gt;dual&lt;/em&gt; (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.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="The bracket"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;The bracket&lt;/h1&gt;