Interior product: 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:~~mbattaglia1~~|~~mbattaglia1~~]] and made on <tt>2011-11-22 15:27:30 UTC</tt>.<br>

: This revision was by author [[User:clumma|clumma]] and made on <tt>2011-11-22 16:05:10 UTC</tt>.<br>

: The original revision id was <tt>~~278250542~~</tt>.<br>

: The original revision id was <tt>278264816</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">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 wedgie product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and V as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).

<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">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 wedgie 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).

For example, suppose W = <<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> and |-1 1 1 1>, then wedging them together gives the bimonzo ||1 1 -1 0 0 0>>. The dot product with W is <<6 -7 -2 -25 -20 15||1 1 -1 0 0 0>>, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0>, |-1 1 1 1>) = 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.

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If we like, we can take the wedge product m∨W from the front by using W(q, s1, s2, s3 ... s_(n-1)) instead of W(s1, s2, s3 ... s_(n-1), q), but this can only lead to a difference in sign.</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>Interior product</title></head><body>The <em>interior product</em> 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 <a class="wiki_link" href="/Wedgies%20and%20Multivals">n-map</a>, 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 wedgie product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and V as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).<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>Interior product</title></head><body>The <em>interior product</em> 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 <a class="wiki_link" href="/Wedgies%20and%20Multivals">n-map</a>, 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 wedgie 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).<br />

<br />

For example, suppose W = &lt;&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&gt; and |-1 1 1 1&gt;, then wedging them together gives the bimonzo ||1 1 -1 0 0 0&gt;&gt;. The dot product with W is &lt;&lt;6 -7 -2 -25 -20 15||1 1 -1 0 0 0&gt;&gt;, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0&gt;, |-1 1 1 1&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.<br />

@@ 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:mbattaglia1|mbattaglia1]] and made on <tt>2011-11-22 15:27:30 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2011-11-22 16:05:10 UTC</tt>.<br>
-: The original revision id was <tt>278250542</tt>.<br>
+: The original revision id was <tt>278264816</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">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 wedgie product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and V as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).
+<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">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 wedgie 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).
 For example, suppose W = &lt;&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&gt; and |-1 1 1 1&gt;, then wedging them together gives the bimonzo ||1 1 -1 0 0 0&gt;&gt;. The dot product with W is &lt;&lt;6 -7 -2 -25 -20 15||1 1 -1 0 0 0&gt;&gt;, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0&gt;, |-1 1 1 1&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.
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 If we like, we can take the wedge product m∨W from the front by using W(q, s1, s2, s3 ... s_(n-1)) instead of W(s1, s2, s3 ... s_(n-1), q), but this can only lead to a difference in sign.</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;Interior product&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;interior product&lt;/em&gt; 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 &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;n-map&lt;/a&gt;, 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 wedgie product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and V as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).&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;Interior product&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;interior product&lt;/em&gt; 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 &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;n-map&lt;/a&gt;, 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 wedgie 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).&lt;br /&gt;
 &lt;br /&gt;
 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;