Graph-theoretic properties of scales: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-28 13:38:38 UTC</tt>.<br>

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[[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; in four dimensions this can be identified as the eight verticies and 24 edges of the[[http://en.wikipedia.org/wiki/16-cell|16-cell]], or hexadecachoron, a four-dimensional regular polytope. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group S2≀S4. Its graph is 6-regular, with every vertex connected to six others (on the 16-cell only opposite verticies fail to connect), 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, consisting of the four edges connecting opposite verticies of the 16-cell. It has 16 maximal cliques, all of which are tetrads, corresponding to the 16 cells of the 16-cell, and the automorphism group acts faithfully on the tetrads. The graph of the maximal cliques 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, the "opposite vertex". The relative complement of each dyadic tetrad is also a dyadic tetrad.

&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; in four dimensions this can be identified as the eight verticies and 24 edges of the&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/16-cell" rel="nofollow"&gt;16-cell&lt;/a&gt;, or hexadecachoron, a four-dimensional regular polytope. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group S2≀S4. Its graph is 6-regular, with every vertex connected to six others (on the 16-cell only opposite verticies fail to connect), 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, consisting of the four edges connecting opposite verticies of the 16-cell. It has 16 maximal cliques, all of which are tetrads, corresponding to the 16 cells of the 16-cell, and the automorphism group acts faithfully on the tetrads. The graph of the maximal cliques 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, the &amp;quot;opposite vertex&amp;quot;. The relative complement of each dyadic tetrad is also a dyadic tetrad.&lt;br /&gt;