Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 359125703 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 359125841 - 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>2012-08-22 01: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-22 01:13:30 UTC</tt>.<br> | ||
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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)/V, where V 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)/V 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. | 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)/V, where V 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)/V 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 | 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 V-1 values of -1 and one of V-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 [[http://mathworld.wolfram.com/GraphDiameter.html|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. | ||
=The Laplace Spectrum= | =The Laplace Spectrum= | ||
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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)/V, where V 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)/V 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 /> | 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)/V, where V 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)/V 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 /> | ||
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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 | 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 V-1 values of -1 and one of V-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 <a class="wiki_link_ext" href="http://mathworld.wolfram.com/GraphDiameter.html" rel="nofollow">diameter</a> 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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="The Laplace Spectrum"></a><!-- ws:end:WikiTextHeadingRule:6 -->The Laplace Spectrum</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="The Laplace Spectrum"></a><!-- ws:end:WikiTextHeadingRule:6 -->The Laplace Spectrum</h1> | ||