Graph-theoretic properties of scales: Difference between revisions

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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>2012-09-06 12:05:56 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-06 17:41:59 UTC</tt>.<br>

: The original revision id was <tt>~~362548036~~</tt>.<br>

: The original revision id was <tt>362653324</tt>.<br>

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

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The 5-limit Fokker block 1-27/25-10/9-6/5-5/4-4/3-36/25-3/2-8/5-5/3-9/5-48/25-2, which has been called the [[duodene_skew|skewed duodene]] or magsyn3, serves as a transversal for the [[novadene|Novadene]], which in 185et is 0, 19, 29, 48, 60, 77, 96, 108, 125, 137, 156, 173, 185 with consonances the 9-limit diamond, 29, 31, 36, 41, 48, 60, 67, 77, 89, 96, 108, 118, 125, 137, 144, 149, 154, 156.

The graph of Novadene has 46 edges between its 12 vertices, and an automorphism group of order 48 isomorphic to the group of the octahedron. The automorphisms permute the four notes with note-number divisible by 3, namely 0, 3, 6 and 9, in all 24 possible ways and the other vertices in 48 ways, so there are two separate orbits. The degrees of 0, 3, 6 and 9 are 9, and of the other verticies 7. It has algebraic, vertex and edge connectivities of 5 ≤ 7 ≤ 7, and a radius and diameter both 2. It has four pentads which are its maximum cliques, and 23 other maximal cliques in the form of tetrads.

It should be noted that the Novadene has a couple of 14/11 thirds which are four cents sharp, and it makes sense to count them among the consonances of Novadene; however, doing so makes the graph less symmetrical.

==Magic[13]==

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The 5-limit Fokker block 1-27/25-10/9-6/5-5/4-4/3-36/25-3/2-8/5-5/3-9/5-48/25-2, which has been called the <a class="wiki_link" href="/duodene_skew">skewed duodene</a> or magsyn3, serves as a transversal for the <a class="wiki_link" href="/novadene">Novadene</a>, which in 185et is 0, 19, 29, 48, 60, 77, 96, 108, 125, 137, 156, 173, 185 with consonances the 9-limit diamond, 29, 31, 36, 41, 48, 60, 67, 77, 89, 96, 108, 118, 125, 137, 144, 149, 154, 156.<br />

<br />

The graph of Novadene has 46 edges between its 12 vertices, and an automorphism group of order 48 isomorphic to the group of the octahedron. The automorphisms permute the four notes with note-number divisible by 3, namely 0, 3, 6 and 9, in all 24 possible ways and the other vertices in 48 ways, so there are two separate orbits. The degrees of 0, 3, 6 and 9 are 9, and of the other verticies 7. It has algebraic, vertex and edge connectivities of 5 ≤ 7 ≤ 7, and a radius and diameter both 2. It has four pentads which are its maximum cliques, and 23 other maximal cliques in the form of tetrads.<br />

<br />

It should be noted that the Novadene has a couple of 14/11 thirds which are four cents sharp, and it makes sense to count them among the consonances of Novadene; however, doing so makes the graph less symmetrical.<br />

<br />

<h2 id="toc17"><a name="Examples-Magic[13]"></a>Magic[13]</h2>

@@ 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>2012-09-06 12:05:56 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-06 17:41:59 UTC</tt>.<br>
-: The original revision id was <tt>362548036</tt>.<br>
+: The original revision id was <tt>362653324</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 124: / Line 124: @@
 The 5-limit Fokker block 1-27/25-10/9-6/5-5/4-4/3-36/25-3/2-8/5-5/3-9/5-48/25-2, which has been called the [[duodene_skew|skewed duodene]] or magsyn3, serves as a transversal for the [[novadene|Novadene]], which in 185et is 0, 19, 29, 48, 60, 77, 96, 108, 125, 137, 156, 173, 185 with consonances the 9-limit diamond, 29, 31, 36, 41, 48, 60, 67, 77, 89, 96, 108, 118, 125, 137, 144, 149, 154, 156.
+The graph of Novadene has 46 edges between its 12 vertices, and an automorphism group of order 48 isomorphic to the group of the octahedron. The automorphisms permute the four notes with note-number divisible by 3, namely 0, 3, 6 and 9, in all 24 possible ways and the other vertices in 48 ways, so there are two separate orbits. The degrees of 0, 3, 6 and 9 are 9, and of the other verticies 7. It has algebraic, vertex and edge connectivities of 5 ≤ 7 ≤ 7, and a radius and diameter both 2. It has four pentads which are its maximum cliques, and 23 other maximal cliques in the form of tetrads.
+It should be noted that the Novadene has a couple of 14/11 thirds which are four cents sharp, and it makes sense to count them among the consonances of Novadene; however, doing so makes the graph less symmetrical.
 ==Magic[13]==
@@ Line 272: / Line 275: @@
 The 5-limit Fokker block 1-27/25-10/9-6/5-5/4-4/3-36/25-3/2-8/5-5/3-9/5-48/25-2, which has been called the &lt;a class="wiki_link" href="/duodene_skew"&gt;skewed duodene&lt;/a&gt; or magsyn3, serves as a transversal for the &lt;a class="wiki_link" href="/novadene"&gt;Novadene&lt;/a&gt;, which in 185et is 0, 19, 29, 48, 60, 77, 96, 108, 125, 137, 156, 173, 185 with consonances the 9-limit diamond, 29, 31, 36, 41, 48, 60, 67, 77, 89, 96, 108, 118, 125, 137, 144, 149, 154, 156.&lt;br /&gt;
 &lt;br /&gt;
+The graph of Novadene has 46 edges between its 12 vertices, and an automorphism group of order 48 isomorphic to the group of the octahedron. The automorphisms permute the four notes with note-number divisible by 3, namely 0, 3, 6 and 9, in all 24 possible ways and the other vertices in 48 ways, so there are two separate orbits. The degrees of 0, 3, 6 and 9 are 9, and of the other verticies 7. It has algebraic, vertex and edge connectivities of 5 ≤ 7 ≤ 7, and a radius and diameter both 2. It has four pentads which are its maximum cliques, and 23 other maximal cliques in the form of tetrads.&lt;br /&gt;
+&lt;br /&gt;
+It should be noted that the Novadene has a couple of 14/11 thirds which are four cents sharp, and it makes sense to count them among the consonances of Novadene; however, doing so makes the graph less symmetrical.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Examples-Magic[13]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Magic[13]&lt;/h2&gt;