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>2011-09-19 01:19:16 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-19 01:22:01 UTC</tt>.

: The original revision id was <tt>~~255486456~~</tt>.

: The original revision id was <tt>255487026</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> 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|; this val tempers out the commas of miracle and ~~also 16~~/15 (or 15~~/14~~), sending all of them to the unison.

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|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison.

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

Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] 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 [<1 1 3 3|, <0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].

Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] 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 an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [<1 1 3 3|, <0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].

* **The [[Tenney-Euclidean Tuning|Frobenius projection map]]**

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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; 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|; this val tempers out the commas of miracle and ~~also 16~~/15 (or 15~~/14~~), sending all of them to the unison.

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|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison.

<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].

<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 an 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].

<ul><li>The <a class="wiki_link" href="/Tenney-Euclidean%20Tuning">Frobenius projection map</a></li></ul>Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection map. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection map, leading to <a class="wiki_link" href="/fractional%20monzos">fractional monzos</a> which are actually the tunings of these intervals in <a class="wiki_link" href="/Fractional%20monzos">Frobenius tuning</a>. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning.

@@ 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>2011-09-19 01:19:16 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-19 01:22:01 UTC</tt>.<br>
-: The original revision id was <tt>255486456</tt>.<br>
+: The original revision id was <tt>255487026</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 14: / Line 14: @@
 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; 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|; this val tempers out the commas of miracle and also 16/15 (or 15/14), sending all of them to the unison.
+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|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison.
 * **[[Normal lists|Normal val lists]]**
-Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] 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].
+Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] 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 an 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].
 * **The [[Tenney-Euclidean Tuning|Frobenius projection map]]**
@@ Line 70: / Line 70: @@
 &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; 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|; this val tempers out the commas of miracle and also 16/15 (or 15/14), sending all of them to the unison. &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|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison. &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;
+&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 an 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;
 &lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;&lt;strong&gt;The &lt;a class="wiki_link" href="/Tenney-Euclidean%20Tuning"&gt;Frobenius projection map&lt;/a&gt;&lt;/strong&gt;&lt;/li&gt;&lt;/ul&gt;Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection map. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection map, leading to &lt;a class="wiki_link" href="/fractional%20monzos"&gt;fractional monzos&lt;/a&gt; which are actually the tunings of these intervals in &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Frobenius tuning&lt;/a&gt;. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning.&lt;br /&gt;