Tenney–Euclidean metrics: 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>2011-01-29 18:51:39 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-29 19:40:41 UTC</tt>.<br>

: The original revision id was <tt>~~197092278~~</tt>.<br>

: The original revision id was <tt>197097588</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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If we want to find the temperamental seminorm T(250/243) of 250/243, we convert it into a monzo as |1 -5 3>. Now we may multiply **P** by this on the left, obtaining the zero vector. Taking the dot product of the zero vector with |1 -5 3> gives zero, and taking the square root of zero we get zero, the temperametal seminorm T(250/243) of 250/243. All of this is telling us that 250/243 is a comma of this temperament, which is 5-limit [[Porcupine family|porcupine]].

Similarly, starting from the monzo |-1 1 0> for 3/2, we may multiply this by **P**, obtaining <-0.8793 0.9957 1.9526|, and taking the dot product of this with |-1 1 0> gives 1.875 with square root 1.3693, which is T(3/2).</pre></div>

Similarly, starting from the monzo |-1 1 0> for 3/2, we may multiply this by **P**, obtaining <-0.8793 0.9957 1.9526|, and taking the dot product of this with |-1 1 0> gives 1.875 with square root 1.3693, which is T(3/2).

We can, however, map the monzos to elements of a rank r abelian group (where r is the rank of the temperament) which abstractly represents the elements of the temperament without regard to tuning, the [[abstract regular temperament]]. If b is a monzo, this mapping is given by bA*. Hence we have |1 -5 3>A* = [0 0] for the interval associated to 250/243, and |-1 1 0>A* = [9 13] for the interval assciated to 3/2. This is the number of steps needed to get to 3/2 in 15et and 22et respectively. We now may obtain a matrix defining the temperamental norm on this abstract temperament by //P// = (AW^2A*)^(-1), which is approximately

[175.3265 -120.0291]

[-120.0291 82.1730]

Using this, we find the temperamental norm of [9 13] to be sqrt([9 13]//P//[9 13]*) ~ sqrt(1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while **P** does not depend on the choice of basis vals for the temperament, //P// does; if we choose [<1 2 3|, <0 -3 -5|] for our basis instead, then 3/2 is represented by [1 -3] and //P// changes coordinates to produce the same final result of temperamental complexity.</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>Tenney-Euclidean metrics</title></head><body><h2 id="toc0"><a name="x-The weighting matrix"></a>The weighting matrix</h2>

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If we want to find the temperamental seminorm T(250/243) of 250/243, we convert it into a monzo as |1 -5 3&gt;. Now we may multiply <strong>P</strong> by this on the left, obtaining the zero vector. Taking the dot product of the zero vector with |1 -5 3&gt; gives zero, and taking the square root of zero we get zero, the temperametal seminorm T(250/243) of 250/243. All of this is telling us that 250/243 is a comma of this temperament, which is 5-limit <a class="wiki_link" href="/Porcupine%20family">porcupine</a>.<br />

<br />

Similarly, starting from the monzo |-1 1 0&gt; for 3/2, we may multiply this by <strong>P</strong>, obtaining &lt;-0.8793 0.9957 1.9526|, and taking the dot product of this with |-1 1 0&gt; gives 1.875 with square root 1.3693, which is T(3/2).</body></html></pre></div>

Similarly, starting from the monzo |-1 1 0&gt; for 3/2, we may multiply this by <strong>P</strong>, obtaining &lt;-0.8793 0.9957 1.9526|, and taking the dot product of this with |-1 1 0&gt; gives 1.875 with square root 1.3693, which is T(3/2).<br />

<br />

We can, however, map the monzos to elements of a rank r abelian group (where r is the rank of the temperament) which abstractly represents the elements of the temperament without regard to tuning, the <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a>. If b is a monzo, this mapping is given by bA*. Hence we have |1 -5 3&gt;A* <h1 id="toc5"><a name="x[0 0] for the interval associated to 250/243, and |-1 1 0&gt;A*"></a> [0 0] for the interval associated to 250/243, and |-1 1 0&gt;A* </h1>

[9 13] for the interval assciated to 3/2. This is the number of steps needed to get to 3/2 in 15et and 22et respectively. We now may obtain a matrix defining the temperamental norm on this abstract temperament by <em>P</em> = (AW^2A*)^(-1), which is approximately<br />

<br />

[175.3265 -120.0291]<br />

[-120.0291 82.1730]<br />

<br />

Using this, we find the temperamental norm of [9 13] to be sqrt([9 13]<em>P</em>[9 13]*) ~ sqrt(1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while <strong>P</strong> does not depend on the choice of basis vals for the temperament, <em>P</em> does; if we choose [&lt;1 2 3|, &lt;0 -3 -5|] for our basis instead, then 3/2 is represented by [1 -3] and <em>P</em> changes coordinates to produce the same final result of temperamental complexity.</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>2011-01-29 18:51:39 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-29 19:40:41 UTC</tt>.<br>
-: The original revision id was <tt>197092278</tt>.<br>
+: The original revision id was <tt>197097588</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 40: / Line 40: @@
 If we want to find the temperamental seminorm T(250/243) of 250/243, we convert it into a monzo as |1 -5 3&gt;. Now we may multiply **P** by this on the left, obtaining the zero vector. Taking the dot product of the zero vector with |1 -5 3&gt; gives zero, and taking the square root of zero we get zero, the temperametal seminorm T(250/243) of 250/243. All of this is telling us that 250/243 is a comma of this temperament, which is 5-limit [[Porcupine family|porcupine]].
-Similarly, starting from the monzo |-1 1 0&gt; for 3/2, we may multiply this by **P**, obtaining &lt;-0.8793 0.9957 1.9526|, and taking the dot product of this with |-1 1 0&gt; gives 1.875 with square root 1.3693, which is T(3/2).</pre></div>
+Similarly, starting from the monzo |-1 1 0&gt; for 3/2, we may multiply this by **P**, obtaining &lt;-0.8793 0.9957 1.9526|, and taking the dot product of this with |-1 1 0&gt; gives 1.875 with square root 1.3693, which is T(3/2).
+We can, however, map the monzos to elements of a rank r abelian group (where r is the rank of the temperament) which abstractly represents the elements of the temperament without regard to tuning, the [[abstract regular temperament]]. If b is a monzo, this mapping is given by bA*. Hence we have |1 -5 3&gt;A* = [0 0] for the interval associated to 250/243, and |-1 1 0&gt;A* = [9 13] for the interval assciated to 3/2. This is the number of steps needed to get to 3/2 in 15et and 22et respectively. We now may obtain a matrix defining the temperamental norm on this abstract temperament by //P// = (AW^2A*)^(-1), which is approximately
+[175.3265 -120.0291]
+[-120.0291 82.1730]
+Using this, we find the temperamental norm of [9 13] to be sqrt([9 13]//P//[9 13]*) ~ sqrt(1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while **P** does not depend on the choice of basis vals for the temperament, //P// does; if we choose [&lt;1 2 3|, &lt;0 -3 -5|] for our basis instead, then 3/2 is represented by [1 -3] and //P// changes coordinates to produce the same final result of temperamental complexity.</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;Tenney-Euclidean metrics&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-The weighting matrix"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;The weighting matrix&lt;/h2&gt;
@@ Line 76: / Line 83: @@
 If we want to find the temperamental seminorm T(250/243) of 250/243, we convert it into a monzo as |1 -5 3&amp;gt;. Now we may multiply &lt;strong&gt;P&lt;/strong&gt; by this on the left, obtaining the zero vector. Taking the dot product of the zero vector with |1 -5 3&amp;gt; gives zero, and taking the square root of zero we get zero, the temperametal seminorm T(250/243) of 250/243. All of this is telling us that 250/243 is a comma of this temperament, which is 5-limit &lt;a class="wiki_link" href="/Porcupine%20family"&gt;porcupine&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-Similarly, starting from the monzo |-1 1 0&amp;gt; for 3/2, we may multiply this by &lt;strong&gt;P&lt;/strong&gt;, obtaining &amp;lt;-0.8793 0.9957 1.9526|, and taking the dot product of this with |-1 1 0&amp;gt; gives 1.875 with square root 1.3693, which is T(3/2).&lt;/body&gt;&lt;/html&gt;</pre></div>
+Similarly, starting from the monzo |-1 1 0&amp;gt; for 3/2, we may multiply this by &lt;strong&gt;P&lt;/strong&gt;, obtaining &amp;lt;-0.8793 0.9957 1.9526|, and taking the dot product of this with |-1 1 0&amp;gt; gives 1.875 with square root 1.3693, which is T(3/2).&lt;br /&gt;
+&lt;br /&gt;
+We can, however, map the monzos to elements of a rank r abelian group (where r is the rank of the temperament) which abstractly represents the elements of the temperament without regard to tuning, the &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt;. If b is a monzo, this mapping is given by bA*. Hence we have |1 -5 3&amp;gt;A*  &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="x[0 0] for the interval associated to 250/243, and |-1 1 0&amp;gt;A*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt; [0 0] for the interval associated to 250/243, and |-1 1 0&amp;gt;A* &lt;/h1&gt;
+ [9 13] for the interval assciated to 3/2. This is the number of steps needed to get to 3/2 in 15et and 22et respectively. We now may obtain a matrix defining the temperamental norm on this abstract temperament by &lt;em&gt;P&lt;/em&gt; = (AW^2A*)^(-1), which is approximately&lt;br /&gt;
+&lt;br /&gt;
+[175.3265 -120.0291]&lt;br /&gt;
+[-120.0291 82.1730]&lt;br /&gt;
+&lt;br /&gt;
+Using this, we find the temperamental norm of [9 13] to be sqrt([9 13]&lt;em&gt;P&lt;/em&gt;[9 13]*) ~ sqrt(1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while &lt;strong&gt;P&lt;/strong&gt; does not depend on the choice of basis vals for the temperament, &lt;em&gt;P&lt;/em&gt; does; if we choose [&amp;lt;1 2 3|, &amp;lt;0 -3 -5|] for our basis instead, then 3/2 is represented by [1 -3] and &lt;em&gt;P&lt;/em&gt; changes coordinates to produce the same final result of temperamental complexity.&lt;/body&gt;&lt;/html&gt;</pre></div>