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:~~guest~~|~~guest~~]] and made on <tt>2012-08-~~22 04~~:33:05 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 02:00:54 UTC</tt>.<br>

: The original revision id was <tt>~~359139245~~</tt>.<br>

: The original revision id was <tt>359382561</tt>.<br>

: The revision comment was: <tt>~~added the unison to the Zarlino scale (hope that's ok)~~</tt><br>

: The revision comment was: <tt></tt><br>

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<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]

=Graph of a scale=

Given a [[periodic scale]], meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by [[Scala]]. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that "5/4" represents {...5/8, 5/4, 5/2, 5, 10 ...} and both "1" and "2" mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 < s < 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class.

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A [[http://en.wikipedia.org/wiki/Clique_%28graph_theory%29|clique]] in a graph is a subgraph such that there is an edge connecting every vertex; that is, the subgraph is a complete graph. In music terms, a clique is a [[dyadic chord]]. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the [[http://en.wikipedia.org/wiki/Clique_problem|clique problem]], and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices.

Among all cliques, the [[http://mathworld.wolfram.com/MaximalClique.html|maximal cliques]], those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A //maximum clique// is a clique of largest size; these are always maximal but the converse does not always hold.

=The Characteristic Polynomial=

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=Examples=

==The Zarlino scale==

The Zarlino scale, or "just diatonic" as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.

[[image:zarlino.png]]

==The diatonic scale (Meantone[7]==

The diatonic scale in 31edo consists of the notes 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of 7edo ) in either the 5 or 7 limits. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.

The genus of the 7-limit diatonic scale is 1, with maximal genus of 4. The connectivities go 3.198 ≤ 4 ≤ 4, and the radius and diameter are both 2.

~~=The Zarlino scale=~~

[[image:diatonic7.gif]]</pre></div>

The Zarlino scale, or "just diatonic" as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.</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"><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> | <a href="#The Automorphism Group">The Automorphism Group</a> | <a href="#Examples">Examples</a> | <~~a href="#The Zarlino scale"~~>~~The Zarlino scale~~</a>

<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> | <a href="#The Automorphism Group">The Automorphism Group</a> | <a href="#Examples">Examples</a>

<h1 id="toc0"><a name="Graph of a scale"></a>Graph of a scale</h1>

<br />

<h1 id="toc0"><a name="Graph of a scale"></a>Graph of a scale</h1>

Given a <a class="wiki_link" href="/periodic%20scale">periodic scale</a>, meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by <a class="wiki_link" href="/Scala">Scala</a>. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that &quot;5/4&quot; represents {...5/8, 5/4, 5/2, 5, 10 ...} and both &quot;1&quot; and &quot;2&quot; mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 &lt; s &lt; 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class.<br />

<br />

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<br />

A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_%28graph_theory%29" rel="nofollow">clique</a> in a graph is a subgraph such that there is an edge connecting every vertex; that is, the subgraph is a complete graph. In music terms, a clique is a <a class="wiki_link" href="/dyadic%20chord">dyadic chord</a>. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_problem" rel="nofollow">clique problem</a>, and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices.<br />

<br />

Among all cliques, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/MaximalClique.html" rel="nofollow">maximal cliques</a>, those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A <em>maximum clique</em> is a clique of largest size; these are always maximal but the converse does not always hold.<br />

<br />

<h1 id="toc2"><a name="The Characteristic Polynomial"></a>The Characteristic Polynomial</h1>

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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 />

<br />

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

<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. 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 />

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<br />

<h1 id="toc6"><a name="Examples"></a>Examples</h1>

~~<br />~~

<h2 id="toc7"><a name="Examples-The Zarlino scale"></a>The Zarlino scale</h2>

<h1 id="toc7"><a name="The Zarlino scale"></a>The Zarlino scale</h1>

The Zarlino scale, or &quot;just diatonic&quot; as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.<br />

The Zarlino scale, or &quot;just diatonic&quot; as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.</body></html></pre></div>

<br />

<img src="/file/view/zarlino.png/359381659/zarlino.png" alt="zarlino.png" title="zarlino.png" /><br />

<br />

<h2 id="toc8"><a name="Examples-The diatonic scale (Meantone[7]"></a>The diatonic scale (Meantone[7]</h2>

The diatonic scale in 31edo consists of the notes 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of 7edo ) in either the 5 or 7 limits. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.<br />

<br />

The genus of the 7-limit diatonic scale is 1, with maximal genus of 4. The connectivities go 3.198 ≤ 4 ≤ 4, and the radius and diameter are both 2.<br />

<br />

<img src="/file/view/diatonic7.gif/359381571/diatonic7.gif" alt="diatonic7.gif" title="diatonic7.gif" /></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:guest|guest]] and made on <tt>2012-08-22 04:33:05 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 02:00:54 UTC</tt>.<br>
-: The original revision id was <tt>359139245</tt>.<br>
+: The original revision id was <tt>359382561</tt>.<br>
-: The revision comment was: <tt>added the unison to the Zarlino scale (hope that's ok)</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>
 <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
 =Graph of a scale=
 Given a [[periodic scale]], meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by [[Scala]]. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that "5/4" represents {...5/8, 5/4, 5/2, 5, 10 ...} and both "1" and "2" mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 &lt; s &lt; 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class.
@@ Line 16: / Line 17: @@
 A [[http://en.wikipedia.org/wiki/Clique_%28graph_theory%29|clique]] in a graph is a subgraph such that there is an edge connecting every vertex; that is, the subgraph is a complete graph. In music terms, a clique is a [[dyadic chord]]. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the [[http://en.wikipedia.org/wiki/Clique_problem|clique problem]], and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices.
+Among all cliques, the [[http://mathworld.wolfram.com/MaximalClique.html|maximal cliques]], those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A //maximum clique// is a clique of largest size; these are always maximal but the converse does not always hold.
 =The Characteristic Polynomial=
@@ Line 54: / Line 57: @@
 =Examples=
+==The Zarlino scale==
+The Zarlino scale, or "just diatonic" as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.
+[[image:zarlino.png]]
+==The diatonic scale (Meantone[7]==
+The diatonic scale in 31edo consists of the notes 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of 7edo ) in either the 5 or 7 limits. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.
+The genus of the 7-limit diatonic scale is 1, with maximal genus of 4. The connectivities go 3.198 ≤ 4 ≤ 4, and the radius and diameter are both 2.
-=The Zarlino scale=
+[[image:diatonic7.gif]]</pre></div>
-The Zarlino scale, or "just diatonic" as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.</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:16:&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:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#Graph of a scale"&gt;Graph of a scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Connectivity"&gt;Connectivity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#The Characteristic Polynomial"&gt;The Characteristic Polynomial&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#The Laplace Spectrum"&gt;The Laplace Spectrum&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#The Genus"&gt;The Genus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#The Automorphism Group"&gt;The Automorphism Group&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Examples"&gt;Examples&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#The Zarlino scale"&gt;The Zarlino scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
+<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:18:&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:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;a href="#Graph of a scale"&gt;Graph of a scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Connectivity"&gt;Connectivity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#The Characteristic Polynomial"&gt;The Characteristic Polynomial&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#The Laplace Spectrum"&gt;The Laplace Spectrum&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#The Genus"&gt;The Genus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#The Automorphism Group"&gt;The Automorphism Group&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Examples"&gt;Examples&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Graph of a scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Graph of a scale&lt;/h1&gt;
+&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Graph of a scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Graph of a scale&lt;/h1&gt;
   Given a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt;, meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by &lt;a class="wiki_link" href="/Scala"&gt;Scala&lt;/a&gt;. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that &amp;quot;5/4&amp;quot; represents {...5/8, 5/4, 5/2, 5, 10 ...} and both &amp;quot;1&amp;quot; and &amp;quot;2&amp;quot; mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 &amp;lt; s &amp;lt; 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class.&lt;br /&gt;
 &lt;br /&gt;
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 &lt;br /&gt;
 A &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_%28graph_theory%29" rel="nofollow"&gt;clique&lt;/a&gt; in a graph is a subgraph such that there is an edge connecting every vertex; that is, the subgraph is a complete graph. In music terms, a clique is a &lt;a class="wiki_link" href="/dyadic%20chord"&gt;dyadic chord&lt;/a&gt;. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_problem" rel="nofollow"&gt;clique problem&lt;/a&gt;, and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices.&lt;br /&gt;
+&lt;br /&gt;
+Among all cliques, the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/MaximalClique.html" rel="nofollow"&gt;maximal cliques&lt;/a&gt;, those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A &lt;em&gt;maximum clique&lt;/em&gt; is a clique of largest size; these are always maximal but the converse does not always hold.&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="The Characteristic Polynomial"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;The Characteristic Polynomial&lt;/h1&gt;
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 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;
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-&lt;!-- ws:start:WikiTextLocalImageRule:26:&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:26 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:29:&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:29 --&gt;&lt;br /&gt;
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 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;
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 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Examples&lt;/h1&gt;
-  &lt;br /&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Examples-The Zarlino scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;The Zarlino scale&lt;/h2&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="The Zarlino scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;The Zarlino scale&lt;/h1&gt;
+The Zarlino scale, or &amp;quot;just diatonic&amp;quot; as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.&lt;br /&gt;
- The Zarlino scale, or &amp;quot;just diatonic&amp;quot; as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:30:&amp;lt;img src=&amp;quot;/file/view/zarlino.png/359381659/zarlino.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/zarlino.png/359381659/zarlino.png" alt="zarlino.png" title="zarlino.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:30 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Examples-The diatonic scale (Meantone[7]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;The diatonic scale (Meantone[7]&lt;/h2&gt;
+The diatonic scale in 31edo consists of the notes 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of 7edo ) in either the 5 or 7 limits. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.&lt;br /&gt;
+&lt;br /&gt;
+The genus of the 7-limit diatonic scale is 1, with maximal genus of 4. The connectivities go 3.198 ≤ 4 ≤ 4, and the radius and diameter are both 2.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:31:&amp;lt;img src=&amp;quot;/file/view/diatonic7.gif/359381571/diatonic7.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/diatonic7.gif/359381571/diatonic7.gif" alt="diatonic7.gif" title="diatonic7.gif" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:31 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>