Graph-theoretic properties of scales: Difference between revisions

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==Star==

[[Star]] is a scale of [[Starling temperaments#Valentine temperament-11-limit|11-limit valentine temperament]], which in [[77edo|77et]] is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads. The graph of the maximal clique is 14 regular.

[[Star]] is a scale of [[Starling temperaments#Valentine temperament-11-limit|11-limit valentine temperament]], which in [[77edo|77et]] is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads. The graph of the maximal clique is 14 regular, with each tetrad linking to all others but its relative complement in the whole scale, that is the set of notes from note 0 to note 7. he relative complement of each dyadic tetrad is also a dyadic tetrad.

[[image:star.png]]

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<br />

<h2 id="toc9"><a name="Examples-Star"></a>Star</h2>

<a class="wiki_link" href="/Star">Star</a> is a scale of <a class="wiki_link" href="/Starling%20temperaments#Valentine temperament-11-limit">11-limit valentine temperament</a>, which in <a class="wiki_link" href="/77edo">77et</a> is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads. The graph of the maximal clique is 14 regular.<br />

<a class="wiki_link" href="/Star">Star</a> is a scale of <a class="wiki_link" href="/Starling%20temperaments#Valentine temperament-11-limit">11-limit valentine temperament</a>, which in <a class="wiki_link" href="/77edo">77et</a> is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads. The graph of the maximal clique is 14 regular, with each tetrad linking to all others but its relative complement in the whole scale, that is the set of notes from note 0 to note 7. he relative complement of each dyadic tetrad is also a dyadic tetrad.<br />

<br />

<img src="/file/view/star.png/359553295/star.png" alt="star.png" title="star.png" /><br />

@@ 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>2012-08-28 00:26:04 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-28 01:02:18 UTC</tt>.<br>
-: The original revision id was <tt>360355686</tt>.<br>
+: The original revision id was <tt>360359928</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 70: / Line 70: @@
 ==Star==
-[[Star]] is a scale of [[Starling temperaments#Valentine temperament-11-limit|11-limit valentine temperament]], which in [[77edo|77et]] is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads. The graph of the maximal clique is 14 regular.
+[[Star]] is a scale of [[Starling temperaments#Valentine temperament-11-limit|11-limit valentine temperament]], which in [[77edo|77et]] is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads. The graph of the maximal clique is 14 regular, with each tetrad linking to all others but its relative complement in the whole scale, that is the set of notes from note 0 to note 7. he relative complement of each dyadic tetrad is also a dyadic tetrad.
 [[image:star.png]]
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Examples-Star"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Star&lt;/h2&gt;
-&lt;a class="wiki_link" href="/Star"&gt;Star&lt;/a&gt; is a scale of &lt;a class="wiki_link" href="/Starling%20temperaments#Valentine temperament-11-limit"&gt;11-limit valentine temperament&lt;/a&gt;, which in &lt;a class="wiki_link" href="/77edo"&gt;77et&lt;/a&gt; is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads. The graph of the maximal clique is 14 regular.&lt;br /&gt;
+&lt;a class="wiki_link" href="/Star"&gt;Star&lt;/a&gt; is a scale of &lt;a class="wiki_link" href="/Starling%20temperaments#Valentine temperament-11-limit"&gt;11-limit valentine temperament&lt;/a&gt;, which in &lt;a class="wiki_link" href="/77edo"&gt;77et&lt;/a&gt; is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads. The graph of the maximal clique is 14 regular, with each tetrad linking to all others but its relative complement in the whole scale, that is the set of notes from note 0 to note 7. he relative complement of each dyadic tetrad is also a dyadic tetrad.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextLocalImageRule:59:&amp;lt;img src=&amp;quot;/file/view/star.png/359553295/star.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/star.png/359553295/star.png" alt="star.png" title="star.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:59 --&gt;&lt;br /&gt;