Regular temperament: 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:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-19 17:07:57 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-19 17:18:59 UTC</tt>.

: The original revision id was <tt>~~203369728~~</tt>.

: The original revision id was <tt>203371252</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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This uses [[http://en.wikipedia.org/wiki/Exterior_algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[http://mathworld.wolfram.com/InteriorProduct.html|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos.

For example, using "v" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1> is <1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0> is also <1 1 3 3|.

For example, using "v" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1> which is <1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0> which is also <1 1 3 3|.

* **[[Normal lists|Normal val lists]]**

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<ul><li>The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></li></ul>This uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">multilinear algebra</a> to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos.

For example, using &quot;v&quot; to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&gt; is &lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&gt; is also &lt;1 1 3 3|.

For example, using &quot;v&quot; to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&gt; which is also &lt;1 1 3 3|.

<ul><li><a class="wiki_link" href="/Normal%20lists">Normal val lists</a></li></ul>Given a list of vals, we may <a class="wiki_link" href="/Saturation">saturate</a> it and reduce it using the <a class="wiki_link" href="/Normal%20lists">Hermite normal form</a> to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [&lt;1 1 3 3|, &lt;0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].

@@ 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>2011-02-19 17:07:57 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-19 17:18:59 UTC</tt>.<br>
-: The original revision id was <tt>203369728</tt>.<br>
+: The original revision id was <tt>203371252</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: @@
 This uses [[http://en.wikipedia.org/wiki/Exterior_algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[http://mathworld.wolfram.com/InteriorProduct.html|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos.
-For example, using "v" to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&gt; is &lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&gt; is also &lt;1 1 3 3|.
+For example, using "v" to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&gt; which is also &lt;1 1 3 3|.
 * **[[Normal lists|Normal val lists]]**
@@ Line 44: / Line 44: @@
 &lt;ul&gt;&lt;li&gt;&lt;strong&gt;The &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;&lt;/strong&gt;&lt;/li&gt;&lt;/ul&gt;This uses &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow"&gt;multilinear algebra&lt;/a&gt; to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow"&gt;interior product&lt;/a&gt; of a &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt; for a p-limit temperament with the p-limit monzos. &lt;br /&gt;
 &lt;br /&gt;
-For example, using &amp;quot;v&amp;quot; to represent the interior product, we have mir = &amp;lt;&amp;lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0&amp;gt; is &amp;lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&amp;gt; is &amp;lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&amp;gt; is also &amp;lt;1 1 3 3|. &lt;br /&gt;
+For example, using &amp;quot;v&amp;quot; to represent the interior product, we have mir = &amp;lt;&amp;lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0&amp;gt; is &amp;lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&amp;gt; which is &amp;lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&amp;gt; which is also &amp;lt;1 1 3 3|. &lt;br /&gt;
 &lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;&lt;strong&gt;&lt;a class="wiki_link" href="/Normal%20lists"&gt;Normal val lists&lt;/a&gt;&lt;/strong&gt;&lt;/li&gt;&lt;/ul&gt;Given a list of vals, we may &lt;a class="wiki_link" href="/Saturation"&gt;saturate&lt;/a&gt; it and reduce it using the &lt;a class="wiki_link" href="/Normal%20lists"&gt;Hermite normal form&lt;/a&gt; to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [&amp;lt;1 1 3 3|, &amp;lt;0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1]. &lt;br /&gt;