Hodge dual
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[[toc|flat]]
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=
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. Then 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=
Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that <U|Vº> = U∧V for every k-multival U.
=Computing the dual=
We can see how to go about computing the dual by looking at the dual of wedge products of basis elements. If we have n basis elements B = {b1, b2, ..., bn}, then if we have a k-fold wedge product of k of these basis elements in order, we may subtract the set of these k elements {bi}, in the sense of set theory subtraction, from B. In other words, if we have a k-fold product of basis elements X in order, we may take the corresponding (n-k)-fold product Y of all the remaining basis elements in order. Then X∧Y will be 1 if X concatenated with Y gives an even permutation, and -1 if it gives on odd permutation. Hence, we may take the (n-i)th element of the k-vector, and this becomes the ith element of the dual if the permutation of the k basis elements in order, concatenated with the remaining (n-k) elements in order, is an even permutation. If it is an odd permutation, then minus the (n-i)th element becomes the ith element of 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>, 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.Original HTML content:
<html><head><title>The dual</title></head><body><!-- ws:start:WikiTextTocRule:8:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- 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> (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 />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="The bracket"></a><!-- ws:end:WikiTextHeadingRule:0 -->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, <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. Then 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 <a class="wiki_link" href="/interior%20product">interior product</a>, but then we must fuss about the sign. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="The dual"></a><!-- ws:end:WikiTextHeadingRule:2 -->The dual</h1>
Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that <U|Vº> = U∧V for every k-multival U.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Computing the dual"></a><!-- ws:end:WikiTextHeadingRule:4 -->Computing the dual</h1>
We can see how to go about computing the dual by looking at the dual of wedge products of basis elements. If we have n basis elements B = {b1, b2, ..., bn}, then if we have a k-fold wedge product of k of these basis elements in order, we may subtract the set of these k elements {bi}, in the sense of set theory subtraction, from B. In other words, if we have a k-fold product of basis elements X in order, we may take the corresponding (n-k)-fold product Y of all the remaining basis elements in order. Then X∧Y will be 1 if X concatenated with Y gives an even permutation, and -1 if it gives on odd permutation. Hence, we may take the (n-i)th element of the k-vector, and this becomes the ith element of the dual if the permutation of the k basis elements in order, concatenated with the remaining (n-k) elements in order, is an even permutation. If it is an odd permutation, then minus the (n-i)th element becomes the ith element of the dual.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><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>∧|-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 <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>