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>2013-01-31 10:23:36 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-31 11:56:34 UTC</tt>.

: The original revision id was <tt>~~403046128~~</tt>.

: The original revision id was <tt>403082070</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 [[Interior product|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos.

For example, using "∨" 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 ∨ |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1> which is <1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0> which is also <1 1 3 3|; <1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces in an additional comma, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.

For example, using "∨" 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 ∨ |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1> which is <1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0> which is also <1 1 3 3|; <1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.

As explained on the [[Interior product#Applications|interior product]] page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q) • V.

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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" href="/Interior%20product">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;∨&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 ∨ |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; &lt;1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces in an additional comma, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.

For example, using &quot;∨&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 ∨ |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; &lt;1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.

As explained on the <a class="wiki_link" href="/Interior%20product#Applications">interior product</a> page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q) • V.

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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>2013-01-31 10:23:36 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-31 11:56:34 UTC</tt>.<br>
-: The original revision id was <tt>403046128</tt>.<br>
+: The original revision id was <tt>403082070</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 13: / Line 13: @@
 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 [[Interior product|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos.
-For example, using "∨" 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 ∨ |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; &lt;1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces in an additional comma, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.
+For example, using "∨" 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 ∨ |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; &lt;1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.
 As explained on the [[Interior product#Applications|interior product]] page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q) • V.
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 &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" href="/Interior%20product"&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;∨&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 ∨ |1 0 0 0&amp;gt; is &amp;lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&amp;gt; which is &amp;lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&amp;gt; which is also &amp;lt;1 1 3 3|; &amp;lt;1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces in an additional comma, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.&lt;br /&gt;
+For example, using &amp;quot;∨&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 ∨ |1 0 0 0&amp;gt; is &amp;lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&amp;gt; which is &amp;lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&amp;gt; which is also &amp;lt;1 1 3 3|; &amp;lt;1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.&lt;br /&gt;
 &lt;br /&gt;
 As explained on the &lt;a class="wiki_link" href="/Interior%20product#Applications"&gt;interior product&lt;/a&gt; page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q) • V.&lt;br /&gt;