Interior product: Difference between revisions

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<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">[[toc|flat]]

The //interior product// is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or [[Wedgies and Multivals|n-map]], a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).

=Applications=  

One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if and only if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.

Another application is the use of the interior product to define the intervals of the [[abstract regular temperament]] given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r primes, where r is the rank of W. Form the matrix [W∨2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product), which can be taken to be an (r-1)-multimonzo. Then for any (r-1)-multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q. It should be noted that V with this property is underdetermined, so that many possible multimonzo vectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [W∨2, W∨3, ... W∨p] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of Meantone∨q, where "Meantone" is the 7-limit wedgie, with |1200+300*log2(5), -1200, 0, 0&gt; gives the value in cents of the [[quarter-comma meantone]] tuning of the interval denoted by Meantone∨q.

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<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;Interior product&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:4:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:4 --&gt;&lt;!-- ws:start:WikiTextTocRule:5: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:5 --&gt;&lt;!-- ws:start:WikiTextTocRule:6: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;

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For example, suppose W = &amp;lt;&amp;lt;6 -7 -2 -25 -20 15||, the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely |1 0 0 0&amp;gt; and |-1 1 1 1&amp;gt;, then wedging them together gives the bimonzo ||1 1 -1 0 0 0&amp;gt;&amp;gt;. The dot product with W is &amp;lt;&amp;lt;6 -7 -2 -25 -20 15||1 1 -1 0 0 0&amp;gt;&amp;gt;, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0&amp;gt;, |-1 1 1 1&amp;gt;) = 1. The fact that the result is ∓1 tells us that 2 and 15/14 can serve as a pair of generators for miracle; if the absolute value of the n-map is N, then the monzos it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament.&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Applications&lt;/h1&gt;

  One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if and only if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.&lt;br /&gt;

Another application is the use of the interior product to define the intervals of the &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt; given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r primes, where r is the rank of W. Form the matrix [W∨2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product), which can be taken to be an (r-1)-multimonzo. Then for any (r-1)-multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q. It should be noted that V with this property is underdetermined, so that many possible multimonzo vectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [W∨2, W∨3, ... W∨p] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of Meantone∨q, where &amp;quot;Meantone&amp;quot; is the 7-limit wedgie, with |1200+300*log2(5), -1200, 0, 0&amp;gt; gives the value in cents of the &lt;a class="wiki_link" href="/quarter-comma%20meantone"&gt;quarter-comma meantone&lt;/a&gt; tuning of the interval denoted by Meantone∨q.&lt;br /&gt;

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