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:genewardsmith|genewardsmith]] and made on <tt>2012-01-02 00:40:25 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-02 01:35:27 UTC</tt>.<br>

: The original revision id was <tt>~~289018761~~</tt>.<br>

: The original revision id was <tt>289020919</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>

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=The bracket=

Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, <W|M>, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product W = 612∧441 = <<18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0>∧|-2 1 -1 1> = ||6 -4 0 -1 3 -2>>. Then <W|M> equals <<18 27 18 1 -22 -34||6 -4 0 -1 3 -2>> equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the [[interior product]], but then we must fuss about the sign.

Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, <W|M>, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product of the 7-limit patent vals 612 and 441, W = 612∧441 = <<18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0>∧|-2 1 -1 1> = ||6 -4 0 -1 3 -2>>. Then <W|M> equals <<18 27 18 1 -22 -34||6 -4 0 -1 3 -2>> equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the [[interior product]], but then we must fuss about the sign.

=The dual=

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<br />

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

Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, &lt;W|M&gt;, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product W = 612∧441 = &lt;&lt;18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0&gt;∧|-2 1 -1 1&gt; = ||6 -4 0 -1 3 -2&gt;&gt;. Then &lt;W|M&gt; equals &lt;&lt;18 27 18 1 -22 -34||6 -4 0 -1 3 -2&gt;&gt; equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the <a class="wiki_link" href="/interior%20product">interior product</a>, but then we must fuss about the sign. <br />

Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, &lt;W|M&gt;, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product of the 7-limit patent vals 612 and 441, W = 612∧441 = &lt;&lt;18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0&gt;∧|-2 1 -1 1&gt; = ||6 -4 0 -1 3 -2&gt;&gt;. Then &lt;W|M&gt; equals &lt;&lt;18 27 18 1 -22 -34||6 -4 0 -1 3 -2&gt;&gt; equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the <a class="wiki_link" href="/interior%20product">interior product</a>, but then we must fuss about the sign. <br />

<br />

<h1 id="toc1"><a name="The dual"></a>The dual</h1>

@@ Line 1: / Line 1: @@
 <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:genewardsmith|genewardsmith]] and made on <tt>2012-01-02 00:40:25 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-02 01:35:27 UTC</tt>.<br>
-: The original revision id was <tt>289018761</tt>.<br>
+: The original revision id was <tt>289020919</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>
@@ Line 11: / Line 11: @@
 =The bracket=
-Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, &lt;W|M&gt;, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product W = 612∧441 = &lt;&lt;18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0&gt;∧|-2 1 -1 1&gt; = ||6 -4 0 -1 3 -2&gt;&gt;. Then &lt;W|M&gt; equals &lt;&lt;18 27 18 1 -22 -34||6 -4 0 -1 3 -2&gt;&gt; equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the [[interior product]], but then we must fuss about the sign.
+Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, &lt;W|M&gt;, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product of the 7-limit patent vals 612 and 441, W = 612∧441 = &lt;&lt;18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0&gt;∧|-2 1 -1 1&gt; = ||6 -4 0 -1 3 -2&gt;&gt;. Then &lt;W|M&gt; equals &lt;&lt;18 27 18 1 -22 -34||6 -4 0 -1 3 -2&gt;&gt; equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the [[interior product]], but then we must fuss about the sign.
 =The dual=
@@ Line 27: / Line 27: @@
 &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;
-Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, &amp;lt;W|M&amp;gt;, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product W = 612∧441 = &amp;lt;&amp;lt;18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0&amp;gt;∧|-2 1 -1 1&amp;gt; = ||6 -4 0 -1 3 -2&amp;gt;&amp;gt;. Then &amp;lt;W|M&amp;gt; equals &amp;lt;&amp;lt;18 27 18 1 -22 -34||6 -4 0 -1 3 -2&amp;gt;&amp;gt; equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the &lt;a class="wiki_link" href="/interior%20product"&gt;interior product&lt;/a&gt;, but then we must fuss about the sign. &lt;br /&gt;
+Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, &amp;lt;W|M&amp;gt;, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product of the 7-limit patent vals 612 and 441, W = 612∧441 = &amp;lt;&amp;lt;18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0&amp;gt;∧|-2 1 -1 1&amp;gt; = ||6 -4 0 -1 3 -2&amp;gt;&amp;gt;. Then &amp;lt;W|M&amp;gt; equals &amp;lt;&amp;lt;18 27 18 1 -22 -34||6 -4 0 -1 3 -2&amp;gt;&amp;gt; equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the &lt;a class="wiki_link" href="/interior%20product"&gt;interior product&lt;/a&gt;, but then we must fuss about the sign. &lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="The dual"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;The dual&lt;/h1&gt;