Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 358515829 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 358524241 - 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>2012-08-18 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-18 19:19:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>358524241</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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The [[http://en.wikipedia.org/wiki/Adjacency_matrix|adjacency matrix]] A of a graph is the square symmetric matrix with zeros on the diagonal, with 1 at the (i, j) place if the ith vertex is connected to the jth vertex, and 0 if it is not. The [[http://en.wikipedia.org/wiki/Characteristic_polynomial|characteristic polynomial]] of a graph is the characteristic polynomial det(xI - A) of its adjacency matrix, and is a property (graph invariant) of the graph alone, without consideration of how the vertices are numbered. The degree n-1 term is 0, the n-2 term is minus the number of edges of the graph and therefore dyads of the scale, and the n-3 term minus twice the number of triads (3-cliques) in the graph and therefore the scale. | The [[http://en.wikipedia.org/wiki/Adjacency_matrix|adjacency matrix]] A of a graph is the square symmetric matrix with zeros on the diagonal, with 1 at the (i, j) place if the ith vertex is connected to the jth vertex, and 0 if it is not. The [[http://en.wikipedia.org/wiki/Characteristic_polynomial|characteristic polynomial]] of a graph is the characteristic polynomial det(xI - A) of its adjacency matrix, and is a property (graph invariant) of the graph alone, without consideration of how the vertices are numbered. The degree n-1 term is 0, the n-2 term is minus the number of edges of the graph and therefore dyads of the scale, and the n-3 term minus twice the number of triads (3-cliques) in the graph and therefore the scale. | ||
These properties can be also expressed in terms of the roots of the characteristic polynomial, which are the eigenvalues of the matrix. These roots are real numbers, some of which may be multiple. The //spectrum// of G is the [[http://en.wikipedia.org/wiki/Multiset|multiset]] of roots, including multipicities, so that some roots may be repeated. From [[http://en.wikipedia.org/wiki/Newton's_identities|Newton's identities]] we can also say that the sum of the squares of the | These properties can be also expressed in terms of the roots of the characteristic polynomial, which are the eigenvalues of the matrix. These roots are real numbers, some of which may be multiple. The //spectrum// of G is the [[http://en.wikipedia.org/wiki/Multiset|multiset]] of roots, including multipicities, so that some roots may be repeated. From [[http://en.wikipedia.org/wiki/Newton's_identities|Newton's identities]] we can also say that the sum of the squares of the spectrum is twice the number of edges of the graph, which means twice the number of dyads of the scale, and the sum of the cubes of the spectrum is six times the number of triads of the scale. In terms of the adjacency matrix A, the number of dyads is tr(A^2)/2 and the number of triads is tr(A^3)/6, where "tr" denotes the trace. Since tr(A^2)/n, where n is the degree (number of notes) is the variance of the spectrum, we can see we prefer the variance to be larger rather than smaller. Also, tr(A^3)/n divided by the 3/2 power of the variance is the skewness of the spectrum, from which we can conclude we want the skewness to skew positive. | ||
The epitome of these properties is found in the complete graph, which in scale terms means every note is in consonant relation with every other--in other words, the scale is a dyadic chord. This has a spectrum with n-1 values of -1 a and one of n-1. The [[http://en.wikipedia.org/wiki/Distance_(graph_theory)|distance]] between two vertices (notes) is the number of edges (consonant intervals) of the shortest path between then, and the diameter of a graph is the length of the greatest distance between two vertices. The diameter of a compete graph is 1, and in general a small diameter is another desirable quality. A graph with diameter d must have at least d+1 distinct values in the spectrum.</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>Graph-theoretic properties of scales</title></head><body><!-- ws:start:WikiTextTocRule:6:&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:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Graph of a scale">Graph of a scale</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Connectivity">Connectivity</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Characteristic polynomial">Characteristic polynomial</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <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>Graph-theoretic properties of scales</title></head><body><!-- ws:start:WikiTextTocRule:6:&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:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Graph of a scale">Graph of a scale</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Connectivity">Connectivity</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Characteristic polynomial">Characteristic polynomial</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | ||
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The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adjacency_matrix" rel="nofollow">adjacency matrix</a> A of a graph is the square symmetric matrix with zeros on the diagonal, with 1 at the (i, j) place if the ith vertex is connected to the jth vertex, and 0 if it is not. The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Characteristic_polynomial" rel="nofollow">characteristic polynomial</a> of a graph is the characteristic polynomial det(xI - A) of its adjacency matrix, and is a property (graph invariant) of the graph alone, without consideration of how the vertices are numbered. The degree n-1 term is 0, the n-2 term is minus the number of edges of the graph and therefore dyads of the scale, and the n-3 term minus twice the number of triads (3-cliques) in the graph and therefore the scale. <br /> | The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adjacency_matrix" rel="nofollow">adjacency matrix</a> A of a graph is the square symmetric matrix with zeros on the diagonal, with 1 at the (i, j) place if the ith vertex is connected to the jth vertex, and 0 if it is not. The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Characteristic_polynomial" rel="nofollow">characteristic polynomial</a> of a graph is the characteristic polynomial det(xI - A) of its adjacency matrix, and is a property (graph invariant) of the graph alone, without consideration of how the vertices are numbered. The degree n-1 term is 0, the n-2 term is minus the number of edges of the graph and therefore dyads of the scale, and the n-3 term minus twice the number of triads (3-cliques) in the graph and therefore the scale. <br /> | ||
<br /> | <br /> | ||
These properties can be also expressed in terms of the roots of the characteristic polynomial, which are the eigenvalues of the matrix. These roots are real numbers, some of which may be multiple. The <em>spectrum</em> of G is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multiset" rel="nofollow">multiset</a> of roots, including multipicities, so that some roots may be repeated. From <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Newton's_identities" rel="nofollow">Newton's identities</a> we can also say that the sum of the squares of the | These properties can be also expressed in terms of the roots of the characteristic polynomial, which are the eigenvalues of the matrix. These roots are real numbers, some of which may be multiple. The <em>spectrum</em> of G is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multiset" rel="nofollow">multiset</a> of roots, including multipicities, so that some roots may be repeated. From <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Newton's_identities" rel="nofollow">Newton's identities</a> we can also say that the sum of the squares of the spectrum is twice the number of edges of the graph, which means twice the number of dyads of the scale, and the sum of the cubes of the spectrum is six times the number of triads of the scale. In terms of the adjacency matrix A, the number of dyads is tr(A^2)/2 and the number of triads is tr(A^3)/6, where &quot;tr&quot; denotes the trace. Since tr(A^2)/n, where n is the degree (number of notes) is the variance of the spectrum, we can see we prefer the variance to be larger rather than smaller. Also, tr(A^3)/n divided by the 3/2 power of the variance is the skewness of the spectrum, from which we can conclude we want the skewness to skew positive.<br /> | ||
<br /> | |||
The epitome of these properties is found in the complete graph, which in scale terms means every note is in consonant relation with every other--in other words, the scale is a dyadic chord. This has a spectrum with n-1 values of -1 a and one of n-1. The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Distance_(graph_theory)" rel="nofollow">distance</a> between two vertices (notes) is the number of edges (consonant intervals) of the shortest path between then, and the diameter of a graph is the length of the greatest distance between two vertices. The diameter of a compete graph is 1, and in general a small diameter is another desirable quality. A graph with diameter d must have at least d+1 distinct values in the spectrum.</body></html></pre></div> | |||