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-~~28 21~~:26:57 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-29 18:51:39 UTC</tt>.<br>

: The original revision id was <tt>~~196978688~~</tt>.<br>

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

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

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==Logflat TE badness==

Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then //logflat badness// is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.</pre></div>

Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then //logflat badness// is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.

==Examples==

Consider the temperament defined by the 5-limit [[Patent val|patent vals]] for 15 and 22 equal. From the vals, we may contruct a 2x3 matrix A = [<15 24 35|, <22 35 51|]. From this we may obtain the matrix **P** as A*(AW^2A*)^(-1)A, approximately

[0.9911 0.1118 -0.1440]

[0.1118 1.1075 1.8086]

[-0.1440 1.8086 3.0624]

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>

<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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<br />

<h2 id="toc3"><a name="x-Logflat TE badness"></a>Logflat TE badness</h2>

Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then <em>logflat badness</em> is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.</body></html></pre></div>

Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then <em>logflat badness</em> is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.<br />

<br />

<h2 id="toc4"><a name="x-Examples"></a>Examples</h2>

Consider the temperament defined by the 5-limit <a class="wiki_link" href="/Patent%20val">patent vals</a> for 15 and 22 equal. From the vals, we may contruct a 2x3 matrix A = [&lt;15 24 35|, &lt;22 35 51|]. From this we may obtain the matrix <strong>P</strong> as A*(AW^2A*)^(-1)A, approximately <br />

<br />

[0.9911 0.1118 -0.1440]<br />

[0.1118 1.1075 1.8086]<br />

[-0.1440 1.8086 3.0624]<br />

<br />

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>

@@ 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>2011-01-28 21:26:57 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-29 18:51:39 UTC</tt>.<br>
-: The original revision id was <tt>196978688</tt>.<br>
+: The original revision id was <tt>197092278</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 29: / Line 29: @@
 ==Logflat TE badness==
-Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then //logflat badness// is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.</pre></div>
+Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then //logflat badness// is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.
+==Examples==
+Consider the temperament defined by the 5-limit [[Patent val|patent vals]] for 15 and 22 equal. From the vals, we may contruct a 2x3 matrix A = [&lt;15 24 35|, &lt;22 35 51|]. From this we may obtain the matrix **P** as A*(AW^2A*)^(-1)A, approximately
+[0.9911 0.1118 -0.1440]
+[0.1118 1.1075 1.8086]
+[-0.1440 1.8086 3.0624]
+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>
 <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;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Logflat TE badness"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Logflat TE badness&lt;/h2&gt;
-Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then &lt;em&gt;logflat badness&lt;/em&gt; is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.&lt;/body&gt;&lt;/html&gt;</pre></div>
+Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then &lt;em&gt;logflat badness&lt;/em&gt; is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Examples&lt;/h2&gt;
+Consider the temperament defined by the 5-limit &lt;a class="wiki_link" href="/Patent%20val"&gt;patent vals&lt;/a&gt; for 15 and 22 equal. From the vals, we may contruct a 2x3 matrix A = [&amp;lt;15 24 35|, &amp;lt;22 35 51|]. From this we may obtain the matrix &lt;strong&gt;P&lt;/strong&gt; as A*(AW^2A*)^(-1)A, approximately &lt;br /&gt;
+&lt;br /&gt;
+[0.9911 0.1118 -0.1440]&lt;br /&gt;
+[0.1118 1.1075 1.8086]&lt;br /&gt;
+[-0.1440 1.8086 3.0624]&lt;br /&gt;
+&lt;br /&gt;
+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>