Tenney–Euclidean metrics: Difference between revisions
Wikispaces>genewardsmith **Imported revision 209335108 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 209336160 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2011-03-10 14: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 14:34:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>209336160</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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Denoting the temperament-defined, or temperamental, 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 //temperamental norm// or //temperamental complexity// of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t//P//t*) where t is the image of a monzo b by t = bA* | Denoting the temperament-defined, or temperamental, 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 //temperamental norm// or //temperamental complexity// of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t//P//t*) where t is the image of a monzo b by t = bA* | ||
= | =The OETES= | ||
Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for **P** in terms of the matrix of monzos B. | Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for **P** in terms of the matrix of monzos B. | ||
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</pre></div> | </pre></div> | ||
<h4>Original HTML content:</h4> | <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><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#The weighting matrix">The weighting matrix</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Temperamental complexity">Temperamental complexity</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="# | <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><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#The weighting matrix">The weighting matrix</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Temperamental complexity">Temperamental complexity</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#The OETES">The OETES</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Logflat TE badness">Logflat TE badness</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | ||
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="The weighting matrix"></a><!-- ws:end:WikiTextHeadingRule:0 -->The weighting matrix</h1> | <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="The weighting matrix"></a><!-- ws:end:WikiTextHeadingRule:0 -->The weighting matrix</h1> | ||
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 /> | 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 /> | ||
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Denoting the temperament-defined, or temperamental, 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 <em>temperamental norm</em> or <em>temperamental complexity</em> of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t<em>P</em>t*) where t is the image of a monzo b by t = bA*<br /> | Denoting the temperament-defined, or temperamental, 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 <em>temperamental norm</em> or <em>temperamental complexity</em> of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t<em>P</em>t*) where t is the image of a monzo b by t = bA*<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name=" | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="The OETES"></a><!-- ws:end:WikiTextHeadingRule:4 -->The OETES</h1> | ||
Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for <strong>P</strong> in terms of the matrix of monzos B.<br /> | Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for <strong>P</strong> in terms of the matrix of monzos B.<br /> | ||
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