Graph-theoretic properties of scales: Difference between revisions

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-20 13:58:54 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-20 14:23:27 UTC</tt>.<br>

: The original revision id was <tt>~~358781535~~</tt>.<br>

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

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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 ~~never~~ the case for graphs of scales, ~~as we have assumed~~ that if an ~~interval~~ s is in the ~~consonance set~~, so is its inversion 2/s. ~~Hence, Aut(G) at least contains~~ an element of order 2, ~~the involution induced by~~ inversion. ~~This, of course, preserves~~ the property of being a dyadic chord~~, which is true of any graph automorphism~~, 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 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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<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 ~~never~~ the case for graphs of scales, ~~as we have assumed~~ that if an ~~interval~~ s is in the ~~consonance set~~, so is its inversion 2/s. ~~Hence, Aut(G) at least contains~~ an element of order 2, ~~the involution induced by~~ inversion. ~~This, of course, preserves~~ the property of being a dyadic chord~~, which is true of any graph automorphism~~, making such things of considerable musical interest.</body></html></pre></div>

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.</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>2012-08-20 13:58:54 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-20 14:23:27 UTC</tt>.<br>
-: The original revision id was <tt>358781535</tt>.<br>
+: The original revision id was <tt>358786233</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 never the case for graphs of scales, as we have assumed that if an interval s is in the consonance set, so is its inversion 2/s. Hence, Aut(G) at least contains an element of order 2, the involution induced by inversion. This, of course, preserves the property of being a dyadic chord, which is true of any graph automorphism, 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 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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 &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 never the case for graphs of scales, as we have assumed that if an interval s is in the consonance set, so is its inversion 2/s. Hence, Aut(G) at least contains an element of order 2, the involution induced by inversion. This, of course, preserves the property of being a dyadic chord, which is true of any graph automorphism, making such things of considerable musical interest.&lt;/body&gt;&lt;/html&gt;</pre></div>
+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;/body&gt;&lt;/html&gt;</pre></div>