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:genewardsmith|genewardsmith]] and made on <tt>2010-10-29 17:56:41 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-29 21:27:21 UTC</tt>.<br>

: The original revision id was <tt>~~174882187~~</tt>.<br>

: The original revision id was <tt>174904281</tt>.<br>

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

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<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;white-space: pre-wrap ! important" class="old-revision-html">Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where a* is the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || <a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.

==The weighting matrix==

Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || <a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.

Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The [[RMS tuning|TOP-RMS]] tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. </pre></div>

==Relative TE complexity==

Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The [[RMS tuning|TOP-RMS]] tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. P is a [[http://en.wikipedia.org/wiki/Positive-definite_matrix|positive semidefinite matrix]], so it defines a [[http://en.wikipedia.org/wiki/Definite_bilinear_form|positive semidefinite bilinear form]]. In terms of weighted monzos m1 and m2, m1Pm2* defines the semidefinite form on weighted monzos, and hence b1W^(-1)PW^(-1)b2* defines a semidefinite form on unweighted monzos, in terms of the matrix W(-1)WA*(AW^2A*)^(-1)AWW^(-1) = A*(AW^2A*)^(-1)A = //P//. From the semidefinite form we obtain and associated [[http://en.wikipedia.org/wiki/Definite_quadratic_form|semidefinite quadratic form]] b//P//b* and from this the [[http://en.wikipedia.org/wiki/Norm_%28mathematics%29|seminorm]] sqrt(b//P//b*).

Denoting this temperament-defined seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a sublattice of the lattice of monzos consisting of the commas of the temperament. The [[http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29|quotient space]] of the full vector space by the commatic subspace such that T(x) = 0 is now a [[http://en.wikipedia.org/wiki/Normed_vector_space|normed vector space]] with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the relative TE complexity of the intervals of the regular temperament.</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>Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val &quot;a&quot; expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where a* is the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || &lt;a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.<br />

<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><br />

<h2 id="toc0"><a name="x-The weighting matrix"></a>The weighting matrix</h2>

Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val &quot;a&quot; expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || &lt;a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.<br />

<br />

<h2 id="toc1"><a name="x-Relative TE complexity"></a>Relative TE complexity</h2>

Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The <a class="wiki_link" href="/RMS%20tuning">TOP-RMS</a> tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. P is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Positive-definite_matrix" rel="nofollow">positive semidefinite matrix</a>, so it defines a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_bilinear_form" rel="nofollow">positive semidefinite bilinear form</a>. In terms of weighted monzos m1 and m2, m1Pm2* defines the semidefinite form on weighted monzos, and hence b1W^(-1)PW^(-1)b2* defines a semidefinite form on unweighted monzos, in terms of the matrix W(-1)WA*(AW^2A*)^(-1)AWW^(-1) = A*(AW^2A*)^(-1)A = <em>P</em>. From the semidefinite form we obtain and associated <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_quadratic_form" rel="nofollow">semidefinite quadratic form</a> b<em>P</em>b* and from this the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Norm_%28mathematics%29" rel="nofollow">seminorm</a> sqrt(b<em>P</em>b*). <br />

<br />

~~Suppose now A is a matrix whose rows are vals defining a p~~-~~limit regular temperament. Then~~ the ~~corresponding weighted matrix is V~~ = AW. The <a class="~~wiki_link~~" href="/~~RMS~~%~~20tuning~~">~~TOP-RMS~~</a> ~~tuning matrix is then V`V, where V` is~~ the ~~pseudoinverse. If~~ the ~~rows of V~~ (~~or equivalently, A~~) ~~are linearly independent, then we have V`~~ = ~~V*(VV*)^(-1)~~, ~~where V* denotes~~ the ~~transpose~~. ~~In terms~~ of ~~vals,~~ the ~~tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW~~.</body></html></pre></div>

Denoting this temperament-defined seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a sublattice of the lattice of monzos consisting of the commas of the temperament. The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29" rel="nofollow">quotient space</a> of the full vector space by the commatic subspace such that T(x) = 0 is now a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">normed vector space</a> with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the relative TE complexity of the intervals of the regular temperament.</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>2010-10-29 17:56:41 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-29 21:27:21 UTC</tt>.<br>
-: The original revision id was <tt>174882187</tt>.<br>
+: The original revision id was <tt>174904281</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>
 <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where a* is the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || &lt;a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.
+<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;white-space: pre-wrap ! important" class="old-revision-html">
+==The weighting matrix==
+Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || &lt;a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.
-Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The [[RMS tuning|TOP-RMS]] tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. </pre></div>
+==Relative TE complexity==
+Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The [[RMS tuning|TOP-RMS]] tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. P is a [[http://en.wikipedia.org/wiki/Positive-definite_matrix|positive semidefinite matrix]], so it defines a [[http://en.wikipedia.org/wiki/Definite_bilinear_form|positive semidefinite bilinear form]]. In terms of weighted monzos m1 and m2, m1Pm2* defines the semidefinite form on weighted monzos, and hence b1W^(-1)PW^(-1)b2* defines a semidefinite form on unweighted monzos, in terms of the matrix W(-1)WA*(AW^2A*)^(-1)AWW^(-1) = A*(AW^2A*)^(-1)A = //P//. From the semidefinite form we obtain and associated [[http://en.wikipedia.org/wiki/Definite_quadratic_form|semidefinite quadratic form]] b//P//b* and from this the [[http://en.wikipedia.org/wiki/Norm_%28mathematics%29|seminorm]] sqrt(b//P//b*).
+Denoting this temperament-defined seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a sublattice of the lattice of monzos consisting of the commas of the temperament. The [[http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29|quotient space]] of the full vector space by the commatic subspace such that T(x) = 0 is now a [[http://en.wikipedia.org/wiki/Normed_vector_space|normed vector space]] with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the relative TE complexity of the intervals of the regular temperament.</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;Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val &amp;quot;a&amp;quot; expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where a* is the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || &amp;lt;a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.&lt;br /&gt;
+<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;br /&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;
+Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val &amp;quot;a&amp;quot; expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || &amp;lt;a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Relative TE complexity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Relative TE complexity&lt;/h2&gt;
+Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The &lt;a class="wiki_link" href="/RMS%20tuning"&gt;TOP-RMS&lt;/a&gt; tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. P is a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Positive-definite_matrix" rel="nofollow"&gt;positive semidefinite matrix&lt;/a&gt;, so it defines a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_bilinear_form" rel="nofollow"&gt;positive semidefinite bilinear form&lt;/a&gt;. In terms of weighted monzos m1 and m2, m1Pm2* defines the semidefinite form on weighted monzos, and hence b1W^(-1)PW^(-1)b2* defines a semidefinite form on unweighted monzos, in terms of the matrix W(-1)WA*(AW^2A*)^(-1)AWW^(-1) = A*(AW^2A*)^(-1)A = &lt;em&gt;P&lt;/em&gt;. From the semidefinite form we obtain and associated &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_quadratic_form" rel="nofollow"&gt;semidefinite quadratic form&lt;/a&gt; b&lt;em&gt;P&lt;/em&gt;b* and from this the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Norm_%28mathematics%29" rel="nofollow"&gt;seminorm&lt;/a&gt; sqrt(b&lt;em&gt;P&lt;/em&gt;b*). &lt;br /&gt;
 &lt;br /&gt;
-Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The &lt;a class="wiki_link" href="/RMS%20tuning"&gt;TOP-RMS&lt;/a&gt; tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW.&lt;/body&gt;&lt;/html&gt;</pre></div>
+Denoting this temperament-defined seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a sublattice of the lattice of monzos consisting of the commas of the temperament. The &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29" rel="nofollow"&gt;quotient space&lt;/a&gt; of the full vector space by the commatic subspace such that T(x) = 0 is now a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow"&gt;normed vector space&lt;/a&gt; with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the relative TE complexity of the intervals of the regular temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>