Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 359382709 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 359385787 - Original comment: ** |
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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 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 | 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. | ||
=Examples= | =Examples= | ||
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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 /> | 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 /> | ||
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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 | 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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:12 -->Examples</h1> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:12 -->Examples</h1> | ||