Graph-theoretic properties of scales
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[[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.
We now may define the **graph of the scale**, which more precisely is the graph G = G(S, C) of the scale S together with the consonance set C. This means G is a (simple) [[http://en.wikipedia.org/wiki/Graph_(mathematics)|graph]] in the sense of [[http://en.wikipedia.org/wiki/Graph_theory|graph theory]]. The vertices of G, V(G), is the set S of pitch classes, and an edge is drawn between two distinct pitch classes r and s if X⋂C ≠ ∅, where X = {x/y|x∊r, y∊s}. This means that there is a relation of consonance, as defined by C, between the pitch classes r and s. It should be noted that we have defined things assuming pitches are given multiplicatively, but we can equally well express them in logarithmic terms as for instance by cents or by steps of an [[EDO]].
=Connectivity=
We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be //connected// if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity ε if it is possible to disconnect the graph by removing ε edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity ν if it is possible to disconnect the graph by removing ν vertices (notes of the scale) but no smaller number will do. The graph is //k-edge-connected// if ε ≥ k, and //k-vertex-connected// if ν ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale.
A [[http://en.wikipedia.org/wiki/Clique_(graph_theory)|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.
=The Characteristic Polynomial=
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 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 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 [[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=
If D is the diagonal matrix [Dij], with Dii being the degree of the ith vertex--that is, the number of edges connecting to that vertex--then L = D - A is called the [[http://en.wikipedia.org/wiki/Laplacian_matrix|Laplace matrix]] of the graph G, and its eigenvalues (roots of its characteristic polynomial) is the //Laplace spectrum//. The matrix L is positive-semidefinite; the Laplace spectrum has at least one zero value, with the other values real and non-negative; the number of zero values in the Laplace spectrum is equal to the number of connected components of G, and so there is just one iff G is connected. The second smallest member λ of the Laplace spectrum, which is therefore positive iff G is connected, is called the [[http://en.wikipedia.org/wiki/Algebraic_connectivity|algebraic connectivity]]. The various kinds of connectivity are related by the inequality λ ≤ ν ≤ ε; that is the algebraic connectivity is less than or equal to the vertex connectivity, which is less than or equal to the edge connectivity. In the other direction, λ ≥ 2(1 - cos(π/V))ε, where V is the number of vertices; consequently λ > (π/V)^2 ε.
We can also relate the diameter d to the algebraic connectivity, since 4/(Vλ) ≤ d ≤ 2((Δ+λ)/4λ)ln(V-1), where Δ is the maximal degree of the vertices of G. Also, if ρ is the mean distance, meaning the average of all of the distances in G, then 2/((V-1)λ) + (V-2)/(2(V-1)) ≤ ρ ≤ (V/(V-1))((Δ+λ)/4λ)ln(V-1). Hence when λ is small, distances will be large.
The sum of the Laplace spectrum, which is tr(L), is twice the number of edges (dyads.) Just as with the ordinary spectrum, the Laplace spectrum has at least d+1 distinct values for a graph of diameter d. The largest value in the Laplace spectrum is less than or equal to n, the degree of the graph, meaning the size of the scale; and is equal iff the complementary graph is disconnected, where the complementary graph is the graph of non-consonant relations, that is, the graph which has an edge between two vertices iff G doesn't.
=The Genus=
A graph is //planar// if it can be drawn on a plane, or equivalently on a sphere, in such a way that no edges cross. Not all graphs can be drawn without edge crossings on a sphere, but they all can be drawn without crossings on a suitable [[http://en.wikipedia.org/wiki/Compact_space|compact]][[http://en.wikipedia.org/wiki/Orientability|orientable]] [[http://en.wikipedia.org/wiki/Surface|surface]] with "donut holes". If a surface has g holes, it is of [[http://en.wikipedia.org/wiki/Genus_(mathematics)|genus]] g, where the sphere (or plane) has genus 0. The minimum number of holes needed to draw the graph is the genus of the graph.
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 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:
[[image:pentad.gif]]
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 is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2
=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.
If the spectrum of the graph contains no repeated values, then Aut(G) is an [[http://en.wikipedia.org/wiki/Elementary_abelian_group|elementary abelian 2-group]], meaning all non-identity elements are of order 2. Hence for the more interesting automorphism groups, including non-abelian groups, we need repeated values in the spectrum, in other words eigenvalues with multiplicity greater than one corresponding to eigenspaces with dimension greater than one.Original HTML content:
<html><head><title>Graph-theoretic properties of scales</title></head><body><!-- ws:start:WikiTextTocRule:12:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Graph of a scale">Graph of a scale</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Connectivity">Connectivity</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#The Characteristic Polynomial">The Characteristic Polynomial</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#The Laplace Spectrum">The Laplace Spectrum</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#The Genus">The Genus</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#The Automorphism Group">The Automorphism Group</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: -->
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<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Graph of a scale"></a><!-- ws:end:WikiTextHeadingRule:0 -->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 "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. <br />
<br />
We now may define the <strong>graph of the scale</strong>, which more precisely is the graph G = G(S, C) of the scale S together with the consonance set C. This means G is a (simple) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_(mathematics)" rel="nofollow">graph</a> in the sense of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_theory" rel="nofollow">graph theory</a>. The vertices of G, V(G), is the set S of pitch classes, and an edge is drawn between two distinct pitch classes r and s if X⋂C ≠ ∅, where X = {x/y|x∊r, y∊s}. This means that there is a relation of consonance, as defined by C, between the pitch classes r and s. It should be noted that we have defined things assuming pitches are given multiplicatively, but we can equally well express them in logarithmic terms as for instance by cents or by steps of an <a class="wiki_link" href="/EDO">EDO</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Connectivity"></a><!-- ws:end:WikiTextHeadingRule:2 -->Connectivity</h1>
We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be <em>connected</em> if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity ε if it is possible to disconnect the graph by removing ε edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity ν if it is possible to disconnect the graph by removing ν vertices (notes of the scale) but no smaller number will do. The graph is <em>k-edge-connected</em> if ε ≥ k, and <em>k-vertex-connected</em> if ν ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale.<br />
<br />
A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_(graph_theory)" 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 />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="The Characteristic Polynomial"></a><!-- ws:end:WikiTextHeadingRule:4 -->The Characteristic Polynomial</h1>
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 />
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 "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.<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 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 <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 />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="The Laplace Spectrum"></a><!-- ws:end:WikiTextHeadingRule:6 -->The Laplace Spectrum</h1>
If D is the diagonal matrix [Dij], with Dii being the degree of the ith vertex--that is, the number of edges connecting to that vertex--then L = D - A is called the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Laplacian_matrix" rel="nofollow">Laplace matrix</a> of the graph G, and its eigenvalues (roots of its characteristic polynomial) is the <em>Laplace spectrum</em>. The matrix L is positive-semidefinite; the Laplace spectrum has at least one zero value, with the other values real and non-negative; the number of zero values in the Laplace spectrum is equal to the number of connected components of G, and so there is just one iff G is connected. The second smallest member λ of the Laplace spectrum, which is therefore positive iff G is connected, is called the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_connectivity" rel="nofollow">algebraic connectivity</a>. The various kinds of connectivity are related by the inequality λ ≤ ν ≤ ε; that is the algebraic connectivity is less than or equal to the vertex connectivity, which is less than or equal to the edge connectivity. In the other direction, λ ≥ 2(1 - cos(π/V))ε, where V is the number of vertices; consequently λ > (π/V)^2 ε.<br />
<br />
We can also relate the diameter d to the algebraic connectivity, since 4/(Vλ) ≤ d ≤ 2((Δ+λ)/4λ)ln(V-1), where Δ is the maximal degree of the vertices of G. Also, if ρ is the mean distance, meaning the average of all of the distances in G, then 2/((V-1)λ) + (V-2)/(2(V-1)) ≤ ρ ≤ (V/(V-1))((Δ+λ)/4λ)ln(V-1). Hence when λ is small, distances will be large.<br />
<br />
The sum of the Laplace spectrum, which is tr(L), is twice the number of edges (dyads.) Just as with the ordinary spectrum, the Laplace spectrum has at least d+1 distinct values for a graph of diameter d. The largest value in the Laplace spectrum is less than or equal to n, the degree of the graph, meaning the size of the scale; and is equal iff the complementary graph is disconnected, where the complementary graph is the graph of non-consonant relations, that is, the graph which has an edge between two vertices iff G doesn't.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="The Genus"></a><!-- ws:end:WikiTextHeadingRule:8 -->The Genus</h1>
A graph is <em>planar</em> if it can be drawn on a plane, or equivalently on a sphere, in such a way that no edges cross. Not all graphs can be drawn without edge crossings on a sphere, but they all can be drawn without crossings on a suitable <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Compact_space" rel="nofollow">compact</a><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Orientability" rel="nofollow">orientable</a> <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Surface" rel="nofollow">surface</a> with "donut holes". If a surface has g holes, it is of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Genus_(mathematics)" rel="nofollow">genus</a> g, where the sphere (or plane) has genus 0. The minimum number of holes needed to draw the graph is the genus of the graph. <br />
<br />
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 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 />
<br />
<!-- ws:start:WikiTextLocalImageRule:20:<img src="/file/view/pentad.gif/358612239/pentad.gif" alt="" title="" /> --><img src="/file/view/pentad.gif/358612239/pentad.gif" alt="pentad.gif" title="pentad.gif" /><!-- ws:end:WikiTextLocalImageRule:20 --><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 />
<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>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 is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="The Automorphism Group"></a><!-- ws:end:WikiTextHeadingRule:10 -->The Automorphism Group</h1>
If A is the adjacency matrix, which is a VxV square matrix, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_automorphism" rel="nofollow">automorphism group</a> Aut(G) of the graph G is given explicitly by the set {P|P⋅A = A⋅P} of VxV <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Permutation_matrix" rel="nofollow">permutation matrices</a> 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.<br />
<br />
If the spectrum of the graph contains no repeated values, then Aut(G) is an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Elementary_abelian_group" rel="nofollow">elementary abelian 2-group</a>, meaning all non-identity elements are of order 2. Hence for the more interesting automorphism groups, including non-abelian groups, we need repeated values in the spectrum, in other words eigenvalues with multiplicity greater than one corresponding to eigenspaces with dimension greater than one.</body></html>