Graph-theoretic properties of scales: Difference between revisions

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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.  

The graph of star 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, it consists 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 32 triads extendable to these tetrads, corresponding to the 32 triangular faces of the 16-cell. 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. The graph is known to be of genus 1, so it can be drawn on a square with opposite edges identified.

The first note of any of the 16 tetrads of star can be either 0 or 1, the second 2 or 3, the third 4 or 5, and the fourth 6 or 7. Any of the resulting 16 possible combinations will not have the forbidden intervals (0, 1), (2, 3), (4, 5), or (6, 7) which are too small to give 11-limit consonances.

&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. &lt;br /&gt;

The graph of star 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, it consists 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 32 triads extendable to these tetrads, corresponding to the 32 triangular faces of the 16-cell. 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. The graph is known to be of genus 1, so it can be drawn on a square with opposite edges identified.&lt;br /&gt;

The first note of any of the 16 tetrads of star can be either 0 or 1, the second 2 or 3, the third 4 or 5, and the fourth 6 or 7. Any of the resulting 16 possible combinations will not have the forbidden intervals (0, 1), (2, 3), (4, 5), or (6, 7) which are too small to give 11-limit consonances.&lt;br /&gt;