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:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-03 09:32:42 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-03 09:34:39 UTC</tt>.<br>

: The original revision id was <tt>~~198357458~~</tt>.<br>

: The original revision id was <tt>198358096</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>

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* **[[http://en.wikipedia.org/wiki/Row_echelon_form|Reduced row echelon form]]**

If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.

For example, if we feed [<22 35 51 62|, <31 49 72 87|, <84 133 195 236|] into a reduced row echelon form routine, we obtain [<1 0 3 1|, <0 1 -3/7 8/7|, <0 0 0 0|]. Stripping off the zero val in the final row, we get E = [<1 0 3 1|, <0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1>, and |-1 -1 0 1>E* = [0 1/7]. Multiply by |1 0 0 0>, the val for 2, and the result is |1 0 0 0>E* = [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.</pre></div>

For example, if we feed [<22 35 51 62|, <31 49 72 87|, <84 133 195 236|] into a reduced row echelon form routine, we obtain [<1 0 3 1|, <0 1 -3/7 8/7|, <0 0 0 0|]. Stripping off the zero val in the final row, we get E = [<1 0 3 1|, <0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1>, and |-1 -1 0 1>E* = [0 1/7]. Multiply by |1 0 0 0>, the val for 2, and the result is |1 0 0 0>E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.</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>Regular temperament</title></head><body>An <em>abstract regular temperament</em> is a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are<br />

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<ul><li><strong><a class="wiki_link" href="/Normal%20lists">Normal comma lists</a></strong></li></ul>The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.<br />

<br />

<ul><li><strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Row_echelon_form" rel="nofollow">Reduced row echelon form</a></strong></li></ul>~~<br />~~

<ul><li><strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Row_echelon_form" rel="nofollow">Reduced row echelon form</a></strong></li></ul>If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.<br />

If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.<br />

<br />

For example, if we feed [&lt;22 35 51 62|, &lt;31 49 72 87|, &lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|, &lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&gt;, and |-1 -1 0 1&gt;E* ~~<h1 id="toc0"><a name~~="x[0 1/7]. Multiply by |1 0 0 0&gt;, the val for 2, and the result is |1 0 0 0&gt;E*~~"></a> [0 1/7]. Multiply by |1 0 0 0&gt;, the val for 2~~, ~~and the result~~ is ~~|1 0 0 0&gt;E* </h1>~~

For example, if we feed [&lt;22 35 51 62|, &lt;31 49 72 87|, &lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|, &lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&gt;, and |-1 -1 0 1&gt;E* = [0 1/7]. Multiply by |1 0 0 0&gt;, the val for 2, and the result is |1 0 0 0&gt;E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.</body></html></pre></div>

[1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.</body></html></pre></div>

@@ 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>2011-02-03 09:32:42 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-03 09:34:39 UTC</tt>.<br>
-: The original revision id was <tt>198357458</tt>.<br>
+: The original revision id was <tt>198358096</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 28: / Line 28: @@
 * **[[http://en.wikipedia.org/wiki/Row_echelon_form|Reduced row echelon form]]**
 If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.
-For example, if we feed [&lt;22 35 51 62|, &lt;31 49 72 87|, &lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|, &lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&gt;, and |-1 -1 0 1&gt;E* = [0 1/7]. Multiply by |1 0 0 0&gt;, the val for 2, and the result is |1 0 0 0&gt;E* = [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.</pre></div>
+For example, if we feed [&lt;22 35 51 62|, &lt;31 49 72 87|, &lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|, &lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&gt;, and |-1 -1 0 1&gt;E* = [0 1/7]. Multiply by |1 0 0 0&gt;, the val for 2, and the result is |1 0 0 0&gt;E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.</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;Regular temperament&lt;/title&gt;&lt;/head&gt;&lt;body&gt;An &lt;em&gt;abstract regular temperament&lt;/em&gt; is a &lt;a class="wiki_link" href="/regular%20temperament"&gt;regular temperament&lt;/a&gt; considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are&lt;br /&gt;
@@ Line 49: / Line 48: @@
 &lt;ul&gt;&lt;li&gt;&lt;strong&gt;&lt;a class="wiki_link" href="/Normal%20lists"&gt;Normal comma lists&lt;/a&gt;&lt;/strong&gt;&lt;/li&gt;&lt;/ul&gt;The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.&lt;br /&gt;
 &lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;&lt;strong&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Row_echelon_form" rel="nofollow"&gt;Reduced row echelon form&lt;/a&gt;&lt;/strong&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;&lt;strong&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Row_echelon_form" rel="nofollow"&gt;Reduced row echelon form&lt;/a&gt;&lt;/strong&gt;&lt;/li&gt;&lt;/ul&gt;If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.&lt;br /&gt;
-If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.&lt;br /&gt;
 &lt;br /&gt;
-For example, if we feed [&amp;lt;22 35 51 62|, &amp;lt;31 49 72 87|, &amp;lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&amp;lt;1 0 3 1|, &amp;lt;0 1 -3/7 8/7|, &amp;lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&amp;lt;1 0 3 1|, &amp;lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&amp;gt;, and |-1 -1 0 1&amp;gt;E*  &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x[0 1/7]. Multiply by |1 0 0 0&amp;gt;, the val for 2, and the result is |1 0 0 0&amp;gt;E*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt; [0 1/7]. Multiply by |1 0 0 0&amp;gt;, the val for 2, and the result is |1 0 0 0&amp;gt;E* &lt;/h1&gt;
+For example, if we feed [&amp;lt;22 35 51 62|, &amp;lt;31 49 72 87|, &amp;lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&amp;lt;1 0 3 1|, &amp;lt;0 1 -3/7 8/7|, &amp;lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&amp;lt;1 0 3 1|, &amp;lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&amp;gt;, and |-1 -1 0 1&amp;gt;E* = [0 1/7]. Multiply by |1 0 0 0&amp;gt;, the val for 2, and the result is |1 0 0 0&amp;gt;E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.&lt;/body&gt;&lt;/html&gt;</pre></div>
- [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.&lt;/body&gt;&lt;/html&gt;</pre></div>