Graph-theoretic properties of scales: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-29 00:32:58 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-29 00:37:54 UTC</tt>.

: The original revision id was <tt>~~360619970~~</tt>.

: The original revision id was <tt>360620542</tt>.

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

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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 two notes of any of the 16 tetrads of star can be either (0, 2), (0, 3), (1, 2) or (1, 3), and the second two notes any of (4, 6), (4, 7), (5, 6) or (5,7). 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.

The first two notes of any of the 16 tetrads of star can be either (0, 2), (0, 3), (1, 2) or (1, 3), and the second two notes any of (4, 6), (4, 7), (5, 6) or (5, 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.

[[image:star.png]]

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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 &quot;opposite vertex&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.

The first two notes of any of the 16 tetrads of star can be either (0, 2), (0, 3), (1, 2) or (1, 3), and the second two notes any of (4, 6), (4, 7), (5, 6) or (5,7). 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.

The first two notes of any of the 16 tetrads of star can be either (0, 2), (0, 3), (1, 2) or (1, 3), and the second two notes any of (4, 6), (4, 7), (5, 6) or (5, 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.

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

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 <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>2012-08-29 00:32:58 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-29 00:37:54 UTC</tt>.<br>
-: The original revision id was <tt>360619970</tt>.<br>
+: The original revision id was <tt>360620542</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 74: / Line 74: @@
 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 two notes of any of the 16 tetrads of star can be either (0, 2), (0, 3), (1, 2) or (1, 3), and the second two notes any of (4, 6), (4, 7), (5, 6) or (5,7). 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.
+The first two notes of any of the 16 tetrads of star can be either (0, 2), (0, 3), (1, 2) or (1, 3), and the second two notes any of (4, 6), (4, 7), (5, 6) or (5, 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.
 [[image:star.png]]
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 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;
 &lt;br /&gt;
-The first two notes of any of the 16 tetrads of star can be either (0, 2), (0, 3), (1, 2) or (1, 3), and the second two notes any of (4, 6), (4, 7), (5, 6) or (5,7). 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;
+The first two notes of any of the 16 tetrads of star can be either (0, 2), (0, 3), (1, 2) or (1, 3), and the second two notes any of (4, 6), (4, 7), (5, 6) or (5, 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;
 &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;