Graph-theoretic properties of scales: Difference between revisions

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<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>2012-08-19 16:10:40 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-19 17:55:53 UTC</tt>.<br>

: The original revision id was <tt>~~358612423~~</tt>.<br>

: The original revision id was <tt>358626879</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>

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Drawing (embedding) the graph on a surface with no crossings would give a nice visual picture of the harmonic relations in a scale. Unfortunately, the problem of determining the genus and finding such an embedding is [[http://en.wikipedia.org/wiki/NP-complete|NP-complete]]. However, finding bounds on the genus is a much easier problem.

A 5-limit scale, meaning a scale composed of 5-limit intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the [[http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif|hexagonal lattice]] of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)

A 5-limit JI scale, meaning a scale composed of 5-limit JI intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the [[http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif|hexagonal lattice]] of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)

A [[dyadic chord]] pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below:

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[[image:pentad.gif]]

The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding.</pre></div>

The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding. On the other hand, a tetrad is of genus 0, since it can be drawn on a sphere as the verticies of a tetrahedron.

If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for [[http://en.wikipedia.org/wiki/Dense_graph|graph density]], 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V>2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) whih has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree of valency is greater than two, then the genus g ≥ 1- V/2 + E/6.

</pre></div>

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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"><html><head><title>Graph-theoretic properties of scales</title></head><body><a href="#Graph of a scale">Graph of a scale</a> | <a href="#Connectivity">Connectivity</a> | <a href="#The Characteristic Polynomial">The Characteristic Polynomial</a> | <a href="#The Laplace Spectrum">The Laplace Spectrum</a> | <a href="#The Genus">The Genus</a>

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Drawing (embedding) the graph on a surface with no crossings would give a nice visual picture of the harmonic relations in a scale. Unfortunately, the problem of determining the genus and finding such an embedding is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/NP-complete" rel="nofollow">NP-complete</a>. However, finding bounds on the genus is a much easier problem.<br />

<br />

A 5-limit scale, meaning a scale composed of 5-limit intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the <a class="wiki_link_ext" href="http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif" rel="nofollow">hexagonal lattice</a> of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)<br />

A 5-limit JI scale, meaning a scale composed of 5-limit JI intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the <a class="wiki_link_ext" href="http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif" rel="nofollow">hexagonal lattice</a> of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)<br />

<br />

A <a class="wiki_link" href="/dyadic%20chord">dyadic chord</a> pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below:<br />

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<img src="/file/view/pentad.gif/358612239/pentad.gif" alt="pentad.gif" title="pentad.gif" /><br />

<br />

The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding.</body></html></pre></div>

The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding. On the other hand, a tetrad is of genus 0, since it can be drawn on a sphere as the verticies of a tetrahedron. <br />

<br />

If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dense_graph" rel="nofollow">graph density</a>, 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V&gt;2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) whih has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree of valency is greater than two, then the genus g ≥ 1- V/2 + E/6.</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>2012-08-19 16:10:40 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-19 17:55:53 UTC</tt>.<br>
-: The original revision id was <tt>358612423</tt>.<br>
+: The original revision id was <tt>358626879</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>
@@ Line 35: / Line 35: @@
 Drawing (embedding) the graph on a surface with no crossings would give a nice visual picture of the harmonic relations in a scale. Unfortunately, the problem of determining the genus and finding such an embedding is [[http://en.wikipedia.org/wiki/NP-complete|NP-complete]]. However, finding bounds on the genus is a much easier problem.
-A 5-limit scale, meaning a scale composed of 5-limit intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the [[http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif|hexagonal lattice]] of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)
+A 5-limit JI scale, meaning a scale composed of 5-limit JI intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the [[http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif|hexagonal lattice]] of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)
 A [[dyadic chord]] pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below:
@@ Line 41: / Line 41: @@
 [[image:pentad.gif]]
-The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding.</pre></div>
+The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding. On the other hand, a tetrad is of genus 0, since it can be drawn on a sphere as the verticies of a tetrahedron.
+If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for [[http://en.wikipedia.org/wiki/Dense_graph|graph density]], 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V&gt;2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) whih has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree of valency is greater than two, then the genus g ≥ 1- V/2 + E/6.
+</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;Graph-theoretic properties of scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&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:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Graph of a scale"&gt;Graph of a scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Connectivity"&gt;Connectivity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#The Characteristic Polynomial"&gt;The Characteristic Polynomial&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#The Laplace Spectrum"&gt;The Laplace Spectrum&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#The Genus"&gt;The Genus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;
@@ Line 72: / Line 76: @@
 Drawing (embedding) the graph on a surface with no crossings would give a nice visual picture of the harmonic relations in a scale. Unfortunately, the problem of determining the genus and finding such an embedding is &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/NP-complete" rel="nofollow"&gt;NP-complete&lt;/a&gt;. However, finding bounds on the genus is a much easier problem.&lt;br /&gt;
 &lt;br /&gt;
-A 5-limit scale, meaning a scale composed of 5-limit intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the &lt;a class="wiki_link_ext" href="http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif" rel="nofollow"&gt;hexagonal lattice&lt;/a&gt; of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)&lt;br /&gt;
+A 5-limit JI scale, meaning a scale composed of 5-limit JI intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the &lt;a class="wiki_link_ext" href="http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif" rel="nofollow"&gt;hexagonal lattice&lt;/a&gt; of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)&lt;br /&gt;
 &lt;br /&gt;
 A &lt;a class="wiki_link" href="/dyadic%20chord"&gt;dyadic chord&lt;/a&gt; pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below:&lt;br /&gt;
@@ Line 78: / Line 82: @@
 &lt;!-- ws:start:WikiTextLocalImageRule:17:&amp;lt;img src=&amp;quot;/file/view/pentad.gif/358612239/pentad.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/pentad.gif/358612239/pentad.gif" alt="pentad.gif" title="pentad.gif" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:17 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding.&lt;/body&gt;&lt;/html&gt;</pre></div>
+The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding. On the other hand, a tetrad is of genus 0, since it can be drawn on a sphere as the verticies of a tetrahedron. &lt;br /&gt;
+&lt;br /&gt;
+If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dense_graph" rel="nofollow"&gt;graph density&lt;/a&gt;, 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V&amp;gt;2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) whih has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree of valency is greater than two, then the genus g ≥ 1- V/2 + E/6.&lt;/body&gt;&lt;/html&gt;</pre></div>