Graph-theoretic properties of scales: Difference between revisions

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==Magic[10]==

Maic[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2.

Maic[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.

Abstractly, the rather large group of automorphisms of order 288 is the direct product of the Klein four-group and the transitive group 6T13 of degree 6, which is the wreath product S3 wr S2. The four-group part acts on the notes from 1 to 4, and is generated by the involutions (1,4) and (2,3), and the 6T13 group, of order 72, acts on notes 5 through 10--or 5 through 9 and 0, if you prefer. It is generated by (5,10)(6,8)(7,9) together with (5,6), (6,7) and (8,9).

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<h2 id="toc12"><a name="Examples-Magic[10]"></a>Magic[10]</h2>

Maic[10], the 10-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, can in <a class="wiki_link" href="/104edo">104et</a> be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2.<br />

Maic[10], the 10-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, can in <a class="wiki_link" href="/104edo">104et</a> be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.<br />

<br />

Abstractly, the rather large group of automorphisms of order 288 is the direct product of the Klein four-group and the transitive group 6T13 of degree 6, which is the wreath product S3 wr S2. The four-group part acts on the notes from 1 to 4, and is generated by the involutions (1,4) and (2,3), and the 6T13 group, of order 72, acts on notes 5 through 10--or 5 through 9 and 0, if you prefer. It is generated by (5,10)(6,8)(7,9) together with (5,6), (6,7) and (8,9).<br />

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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>2012-08-26 21:34:09 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-26 21:41:09 UTC</tt>.<br>
-: The original revision id was <tt>360079757</tt>.<br>
+: The original revision id was <tt>360081091</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 89: / Line 89: @@
 ==Magic[10]==
-Maic[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2.
+Maic[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.
 Abstractly, the rather large group of automorphisms of order 288 is the direct product of the Klein four-group and the transitive group 6T13 of degree 6, which is the wreath product S3 wr S2. The four-group part acts on the notes from 1 to 4, and is generated by the involutions (1,4) and (2,3), and the 6T13 group, of order 72, acts on notes 5 through 10--or 5 through 9 and 0, if you prefer. It is generated by (5,10)(6,8)(7,9) together with (5,6), (6,7) and (8,9).
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Examples-Magic[10]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Magic[10]&lt;/h2&gt;
-Maic[10], the 10-note MOS of &lt;a class="wiki_link" href="/Magic%20family#Magic-11-limit"&gt;magic temperament&lt;/a&gt;, can in &lt;a class="wiki_link" href="/104edo"&gt;104et&lt;/a&gt; be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2.&lt;br /&gt;
+Maic[10], the 10-note MOS of &lt;a class="wiki_link" href="/Magic%20family#Magic-11-limit"&gt;magic temperament&lt;/a&gt;, can in &lt;a class="wiki_link" href="/104edo"&gt;104et&lt;/a&gt; be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.&lt;br /&gt;
 &lt;br /&gt;
 Abstractly, the rather large group of automorphisms of order 288 is the direct product of the Klein four-group and the transitive group 6T13 of degree 6, which is the wreath product S3 wr S2. The four-group part acts on the notes from 1 to 4, and is generated by the involutions (1,4) and (2,3), and the 6T13 group, of order 72, acts on notes 5 through 10--or 5 through 9 and 0, if you prefer. It is generated by (5,10)(6,8)(7,9) together with (5,6), (6,7) and (8,9).&lt;br /&gt;