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>2013-08-17 00:38:21 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-08-17 00:43:36 UTC</tt>.

: The original revision id was <tt>~~445198348~~</tt>.

: The original revision id was <tt>445198698</tt>.

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Like Gypsy, [[marvel11max7a]] is a [[consonant class scale]] for marvel, based on the fact that four 5/4s, two 6/5s and one 8/7 come to four times 225/224, and these are approximately the intervals in the second consonance class, in the order 5/4, 6/5, 5/4, 6/5, 8/7, 5/4, 5/4, leading to a scale which is the marvel tempering of 9/8, 5/4, 32/25, 3/2, 8/5, 15/8, 2. In 166et, this becomes 0, 28, 53, 60, 97, 113, 150, 166, with 11-limit consonance set {21, 23, 25, 28, 32, 37, 44, 48, 53, 58, 60, 69, 76, 81, 85, 90, 97, 106, 108, 113, 118, 122, 129, 134, 138, 141, 143, 145, 166}.

The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other ~~netes~~. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6).

The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other notes. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6).

The faces of the octahdron, [0, 2, 4], [0, 2, 5], [0, 3, 4], [0, 3, 5], [2, 4, 6], [2, 5, 6], [3, 4, 6], [3, 5, 6], are of course triads, and each of them can be extended to a tetrad by adding note 1, leading to the eight maximal cliques of the scale. The connectivities of the scale are 5 ≤ 5 ≤ 5, which is maximal among all seven-note marvel tempered scales save for the fact that the inversion of marvel11max7a, which is [[marvel11max7b]], has of course an isomorphic graph. The count of dyads and triads is also maximal. The genus of the graph is 1.

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Like Gypsy, <a class="wiki_link" href="/marvel11max7a">marvel11max7a</a> is a <a class="wiki_link" href="/consonant%20class%20scale">consonant class scale</a> for marvel, based on the fact that four 5/4s, two 6/5s and one 8/7 come to four times 225/224, and these are approximately the intervals in the second consonance class, in the order 5/4, 6/5, 5/4, 6/5, 8/7, 5/4, 5/4, leading to a scale which is the marvel tempering of 9/8, 5/4, 32/25, 3/2, 8/5, 15/8, 2. In 166et, this becomes 0, 28, 53, 60, 97, 113, 150, 166, with 11-limit consonance set {21, 23, 25, 28, 32, 37, 44, 48, 53, 58, 60, 69, 76, 81, 85, 90, 97, 106, 108, 113, 118, 122, 129, 134, 138, 141, 143, 145, 166}.

The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other ~~netes~~. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6).

The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other notes. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6).

The faces of the octahdron, [0, 2, 4], [0, 2, 5], [0, 3, 4], [0, 3, 5], [2, 4, 6], [2, 5, 6], [3, 4, 6], [3, 5, 6], are of course triads, and each of them can be extended to a tetrad by adding note 1, leading to the eight maximal cliques of the scale. The connectivities of the scale are 5 ≤ 5 ≤ 5, which is maximal among all seven-note marvel tempered scales save for the fact that the inversion of marvel11max7a, which is <a class="wiki_link" href="/marvel11max7b">marvel11max7b</a>, has of course an isomorphic graph. The count of dyads and triads is also maximal. The genus of the graph is 1.

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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>2013-08-17 00:38:21 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-08-17 00:43:36 UTC</tt>.<br>
-: The original revision id was <tt>445198348</tt>.<br>
+: The original revision id was <tt>445198698</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 80: / Line 80: @@
 Like Gypsy, [[marvel11max7a]] is a [[consonant class scale]] for marvel, based on the fact that four 5/4s, two 6/5s and one 8/7 come to four times 225/224, and these are approximately the intervals in the second consonance class, in the order 5/4, 6/5, 5/4, 6/5, 8/7, 5/4, 5/4, leading to a scale which is the marvel tempering of 9/8, 5/4, 32/25, 3/2, 8/5, 15/8, 2. In 166et, this becomes 0, 28, 53, 60, 97, 113, 150, 166, with 11-limit consonance set {21, 23, 25, 28, 32, 37, 44, 48, 53, 58, 60, 69, 76, 81, 85, 90, 97, 106, 108, 113, 118, 122, 129, 134, 138, 141, 143, 145, 166}.
-The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other netes. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6).
+The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other notes. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6).
 The faces of the octahdron, [0, 2, 4], [0, 2, 5], [0, 3, 4], [0, 3, 5], [2, 4, 6], [2, 5, 6], [3, 4, 6], [3, 5, 6], are of course triads, and each of them can be extended to a tetrad by adding note 1, leading to the eight maximal cliques of the scale. The connectivities of the scale are 5 ≤ 5 ≤ 5, which is maximal among all seven-note marvel tempered scales save for the fact that the inversion of marvel11max7a, which is [[marvel11max7b]], has of course an isomorphic graph. The count of dyads and triads is also maximal. The genus of the graph is 1.
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 Like Gypsy, &lt;a class="wiki_link" href="/marvel11max7a"&gt;marvel11max7a&lt;/a&gt; is a &lt;a class="wiki_link" href="/consonant%20class%20scale"&gt;consonant class scale&lt;/a&gt; for marvel, based on the fact that four 5/4s, two 6/5s and one 8/7 come to four times 225/224, and these are approximately the intervals in the second consonance class, in the order 5/4, 6/5, 5/4, 6/5, 8/7, 5/4, 5/4, leading to a scale which is the marvel tempering of 9/8, 5/4, 32/25, 3/2, 8/5, 15/8, 2. In 166et, this becomes 0, 28, 53, 60, 97, 113, 150, 166, with 11-limit consonance set {21, 23, 25, 28, 32, 37, 44, 48, 53, 58, 60, 69, 76, 81, 85, 90, 97, 106, 108, 113, 118, 122, 129, 134, 138, 141, 143, 145, 166}. &lt;br /&gt;
 &lt;br /&gt;
-The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other netes. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6).&lt;br /&gt;
+The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other notes. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6).&lt;br /&gt;
 &lt;br /&gt;
 The faces of the octahdron, [0, 2, 4], [0, 2, 5], [0, 3, 4], [0, 3, 5], [2, 4, 6], [2, 5, 6], [3, 4, 6], [3, 5, 6], are of course triads, and each of them can be extended to a tetrad by adding note 1, leading to the eight maximal cliques of the scale. The connectivities of the scale are 5 ≤ 5 ≤ 5, which is maximal among all seven-note marvel tempered scales save for the fact that the inversion of marvel11max7a, which is &lt;a class="wiki_link" href="/marvel11max7b"&gt;marvel11max7b&lt;/a&gt;, has of course an isomorphic graph. The count of dyads and triads is also maximal. The genus of the graph is 1.&lt;br /&gt;