Tenney–Euclidean metrics: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:~~mbattaglia1~~|~~mbattaglia1~~]] and made on <tt>~~2012~~-07-~~27 21~~:04:51 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-23 20:01:42 UTC</tt>.<br>

: The original revision id was <tt>~~355130806~~</tt>.<br>

: The original revision id was <tt>471702798</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.

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 n is π(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/(n-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

Consider the temperament defined by the 5-limit [[Patent val|patent vals]] for 15 and 22 equal. From the vals, we may construct 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]

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If instead we want the OETES, we may remove the first row of [<1 2 3|, <0 -3 -5|], leaving just [<0 -3 -5|]. If we now call this 1x3 matrix A, then (AW^2A*)^(-1) is a 1x1 matrix; in effect a scalar, with value [<0.1215588|]. Multiplying a monzo b times A* gives a 1x1 matrix bA* whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of steps.

For a more substantial example we need to consider at least a rank three temperament, so let us turn to marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0>, |-5 2 2 -1>]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0>, |-5 2p3 2p5 -p7>], and P = I - M`M = [|1 0 0 0>, |0 4p5^2+p7^2 -4p3p5 2p3p7>/H,

For a more substantial example we need to consider at least a rank three temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0>, |-5 2 2 -1>]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0>, |-5 2p3 2p5 -p7>], and P = I - M`M = [|1 0 0 0>, |0 4p5^2+p7^2 -4p3p5 2p3p7>/H,

|0 -4p3p5 4p3^2+p7^2 2p5p7>/H, |0 2p3p7 2p5p7 4(p3^2+p5^2)>/H], where H = 4p3^2+4p5^2+p7^2. On the other hand, we may start from the normal val list for the temperament, which is [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]. Removing the first row gives [<0 1 0 2|, <0 0 1 2], and val weighting this gives C = [<0 1/p3 0 2/p7|, <0 0 1/p5 2/p7|]. Then P = C`C is precisely the same matrix we obtained before.

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<br />

<h1 id="toc3"><a name="Logflat TE badness"></a>Logflat TE badness</h1>

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 />

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 n is π(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/(n-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 />

<h1 id="toc4"><a name="Examples"></a>Examples</h1>

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 />

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 construct 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 />

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If instead we want the OETES, we may remove the first row of [&lt;1 2 3|, &lt;0 -3 -5|], leaving just [&lt;0 -3 -5|]. If we now call this 1x3 matrix A, then (AW^2A*)^(-1) is a 1x1 matrix; in effect a scalar, with value [&lt;0.1215588|]. Multiplying a monzo b times A* gives a 1x1 matrix bA* whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of steps.<br />

<br />

For a more substantial example we need to consider at least a rank three temperament, so let us turn to marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0&gt;, |-5 2 2 -1&gt;]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0&gt;, |-5 2p3 2p5 -p7&gt;], and P = I - M`M = [|1 0 0 0&gt;, |0 4p5^2+p7^2 -4p3p5 2p3p7&gt;/H,<br />

For a more substantial example we need to consider at least a rank three temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0&gt;, |-5 2 2 -1&gt;]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0&gt;, |-5 2p3 2p5 -p7&gt;], and P = I - M`M = [|1 0 0 0&gt;, |0 4p5^2+p7^2 -4p3p5 2p3p7&gt;/H,<br />

|0 -4p3p5 4p3^2+p7^2 2p5p7&gt;/H, |0 2p3p7 2p5p7 4(p3^2+p5^2)&gt;/H], where H = 4p3^2+4p5^2+p7^2. On the other hand, we may start from the normal val list for the temperament, which is [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]. Removing the first row gives [&lt;0 1 0 2|, &lt;0 0 1 2], and val weighting this gives C = [&lt;0 1/p3 0 2/p7|, &lt;0 0 1/p5 2/p7|]. Then P = C`C is precisely the same matrix we obtained before.<br />

<br />

@@ 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:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-27 21:04:51 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-23 20:01:42 UTC</tt>.<br>
-: The original revision id was <tt>355130806</tt>.<br>
+: The original revision id was <tt>471702798</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 23: / Line 23: @@
 =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.
+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 n is π(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/(n-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
+Consider the temperament defined by the 5-limit [[Patent val|patent vals]] for 15 and 22 equal. From the vals, we may construct 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]
@@ Line 45: / Line 45: @@
 If instead we want the OETES, we may remove the first row of [&lt;1 2 3|, &lt;0 -3 -5|], leaving just [&lt;0 -3 -5|]. If we now call this 1x3 matrix A, then (AW^2A*)^(-1) is a 1x1 matrix; in effect a scalar, with value [&lt;0.1215588|]. Multiplying a monzo b times A* gives a 1x1 matrix bA* whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of steps.
-For a more substantial example we need to consider at least a rank three temperament, so let us turn to marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0&gt;, |-5 2 2 -1&gt;]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0&gt;, |-5 2p3 2p5 -p7&gt;], and P = I - M`M = [|1 0 0 0&gt;, |0 4p5^2+p7^2 -4p3p5 2p3p7&gt;/H,
+For a more substantial example we need to consider at least a rank three temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0&gt;, |-5 2 2 -1&gt;]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0&gt;, |-5 2p3 2p5 -p7&gt;], and P = I - M`M = [|1 0 0 0&gt;, |0 4p5^2+p7^2 -4p3p5 2p3p7&gt;/H,
 |0 -4p3p5 4p3^2+p7^2 2p5p7&gt;/H, |0 2p3p7 2p5p7 4(p3^2+p5^2)&gt;/H], where H = 4p3^2+4p5^2+p7^2. On the other hand, we may start from the normal val list for the temperament, which is [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]. Removing the first row gives [&lt;0 1 0 2|, &lt;0 0 1 2], and val weighting this gives C = [&lt;0 1/p3 0 2/p7|, &lt;0 0 1/p5 2/p7|]. Then P = C`C is precisely the same matrix we obtained before.
@@ Line 69: / Line 69: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Logflat TE badness"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Logflat TE badness&lt;/h1&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;br /&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 n is π(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/(n-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;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Examples&lt;/h1&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;
+  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 construct 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;
@@ Line 91: / Line 91: @@
 If instead we want the OETES, we may remove the first row of [&amp;lt;1 2 3|, &amp;lt;0 -3 -5|], leaving just [&amp;lt;0 -3 -5|]. If we now call this 1x3 matrix A, then (AW^2A*)^(-1) is a 1x1 matrix; in effect a scalar, with value [&amp;lt;0.1215588|]. Multiplying a monzo b times A* gives a 1x1 matrix bA* whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of steps.&lt;br /&gt;
 &lt;br /&gt;
-For a more substantial example we need to consider at least a rank three temperament, so let us turn to marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0&amp;gt;, |-5 2 2 -1&amp;gt;]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0&amp;gt;, |-5 2p3 2p5 -p7&amp;gt;], and P = I - M`M = [|1 0 0 0&amp;gt;, |0 4p5^2+p7^2 -4p3p5 2p3p7&amp;gt;/H,&lt;br /&gt;
+For a more substantial example we need to consider at least a rank three temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0&amp;gt;, |-5 2 2 -1&amp;gt;]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0&amp;gt;, |-5 2p3 2p5 -p7&amp;gt;], and P = I - M`M = [|1 0 0 0&amp;gt;, |0 4p5^2+p7^2 -4p3p5 2p3p7&amp;gt;/H,&lt;br /&gt;
 |0 -4p3p5 4p3^2+p7^2 2p5p7&amp;gt;/H, |0 2p3p7 2p5p7 4(p3^2+p5^2)&amp;gt;/H], where H = 4p3^2+4p5^2+p7^2. On the other hand, we may start from the normal val list for the temperament, which is [&amp;lt;1 0 0 -5|, &amp;lt;0 1 0 2|, &amp;lt;0 0 1 2|]. Removing the first row gives [&amp;lt;0 1 0 2|, &amp;lt;0 0 1 2], and val weighting this gives C = [&amp;lt;0 1/p3 0 2/p7|, &amp;lt;0 0 1/p5 2/p7|]. Then P = C`C is precisely the same matrix we obtained before.&lt;br /&gt;
 &lt;br /&gt;