Graph-theoretic properties of scales: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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=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 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 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.</pre></div>

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<br />

<h1 id="toc5"><a name="The Automorphism Group"></a>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 />

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 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></pre></div>

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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>2012-08-20 14:53:20 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-20 14:55:05 UTC</tt>.<br>
-: The original revision id was <tt>358793097</tt>.<br>
+: The original revision id was <tt>358793441</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 48: / Line 48: @@
 =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 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 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.</pre></div>
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 &lt;br /&gt;
 &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;
-If A is the adjacency matrix, which is a VxV square matrix, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_automorphism" rel="nofollow"&gt;automorphism group&lt;/a&gt; Aut(G) of the graph G is given explicitly by the set {P|P⋅A = A⋅P} of VxV &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Permutation_matrix" rel="nofollow"&gt;permutation matrices&lt;/a&gt; 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.&lt;br /&gt;
+If A is the adjacency matrix, which is a VxV square matrix, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_automorphism" rel="nofollow"&gt;automorphism group&lt;/a&gt; Aut(G) of the graph G is given explicitly by the set {P|P⋅A = A⋅P} of VxV &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Permutation_matrix" rel="nofollow"&gt;permutation matrices&lt;/a&gt; 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 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.&lt;br /&gt;
 &lt;br /&gt;
 If the spectrum of the graph contains no repeated values, then Aut(G) is an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Elementary_abelian_group" rel="nofollow"&gt;elementary abelian 2-group&lt;/a&gt;, 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.&lt;/body&gt;&lt;/html&gt;</pre></div>