Graph-theoretic properties of scales: Difference between revisions

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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)/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 Laplace Spectrum=

=The Automorphism Group=

If A is the adjacency matrix, which is a VxV square matrix, the [[http://en.wikipedia.org/wiki/Graph_automorphism|automorphism group]] Aut(G) of the graph G is given explicitly by the set {P|P⋅A = A⋅P} of VxV [[http://en.wikipedia.org/wiki/Permutation_matrix|permutation matrices]] which commute with A. Equivalently, it is the set {P|A = P⋅A⋅P^(-1)}, where A is identical to itself under a similarity transformation. For most graphs, the automorphism group is trivial; however this is often not the case for graphs of scales. For instance, a symmetric scale such that if an s is in the scale, then so is its octave inversion 2/s will have an element of order 2 in its automorphism group. A MOS will always have an element of order 2, resulting from some composition of inversion and translation. Graph automorphisms such as these preserve the property of being a dyadic chord, making such things of considerable musical interest.

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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Graph-theoretic properties of scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Graph of a scale"&gt;Graph of a scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Connectivity"&gt;Connectivity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#The Characteristic Polynomial"&gt;The Characteristic Polynomial&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#The Laplace Spectrum"&gt;The Laplace Spectrum&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#The Genus"&gt;The Genus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#The Automorphism Group"&gt;The Automorphism Group&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;

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 &lt;em&gt;spectrum&lt;/em&gt; of G is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multiset" rel="nofollow"&gt;multiset&lt;/a&gt; of roots, including multipicities, so that some roots may be repeated. From &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Newton's_identities" rel="nofollow"&gt;Newton's identities&lt;/a&gt; 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 &amp;quot;tr&amp;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.&lt;br /&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="The Laplace Spectrum"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;The Laplace Spectrum&lt;/h1&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="The Automorphism Group"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;The Automorphism Group&lt;/h1&gt;