Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 402164634 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 402209206 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2013-01-28 21:21:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>402209206</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. Class 2 contains the three thirds, 7/6, 6/5 and 5/4. | The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. Class 2 contains the three thirds, 7/6, 6/5 and 5/4. | ||
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//[[http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3|Benny]]// | //[[http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3|Benny]]// | ||
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The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. Class 2 contains the three thirds, 7/6, 6/5 and 5/4.<br /> | The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. Class 2 contains the three thirds, 7/6, 6/5 and 5/4.<br /> | ||
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<em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3" rel="nofollow">Benny</a></em><br /> | <em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3" rel="nofollow">Benny</a></em><br /> | ||
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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.<br /> | 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.<br /> | ||
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<em><a class="wiki_link_ext" href="http://micro.soonlabel.com/star/20120830-77et-star.mp3" rel="nofollow">77et Star</a></em> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> | <em><a class="wiki_link_ext" href="http://micro.soonlabel.com/star/20120830-77et-star.mp3" rel="nofollow">77et Star</a></em> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> | ||
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The graph has twelve maximal cliques, all tetrads, of which four connect with all of the other tetrads, and eight connect with all but one. It has two vertices of degree five and six of degree six, with connectivities 4.586 ≤ 5 ≤ 5, and radius and diameter both 2.<br /> | The graph has twelve maximal cliques, all tetrads, of which four connect with all of the other tetrads, and eight connect with all but one. It has two vertices of degree five and six of degree six, with connectivities 4.586 ≤ 5 ≤ 5, and radius and diameter both 2.<br /> | ||
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<em><a class="wiki_link_ext" href="http://archive.org/download/HighOktoneElgar/oktelg.mp3" rel="nofollow">High Oktone Elgar</a></em><br /> | <em><a class="wiki_link_ext" href="http://archive.org/download/HighOktoneElgar/oktelg.mp3" rel="nofollow">High Oktone Elgar</a></em><br /> | ||
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The graph has 16 maximal cliques, eight tetrads and eight pentads. All of the tetrads contain note 0, and all of the pentads notes 1 and 8. All three connectivites equal 6, the radius and diameter are both 2, and the graph complement is disconnected.<br /> | The graph has 16 maximal cliques, eight tetrads and eight pentads. All of the tetrads contain note 0, and all of the pentads notes 1 and 8. All three connectivites equal 6, the radius and diameter are both 2, and the graph complement is disconnected.<br /> | ||
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<em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Grood/Mountain_Village_TarkanGrood.mp3" rel="nofollow">Mountain Village</a></em> by <a class="wiki_link" href="/Tarkan%20Grood">Tarkan Grood</a><br /> | <em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Grood/Mountain_Village_TarkanGrood.mp3" rel="nofollow">Mountain Village</a></em> by <a class="wiki_link" href="/Tarkan%20Grood">Tarkan Grood</a><br /> | ||
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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 ≀ 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 /> | 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 ≀ 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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<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Ten note scales-The dekany"></a><!-- ws:end:WikiTextHeadingRule:38 -->The dekany</h2> | <!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Ten note scales-The dekany"></a><!-- ws:end:WikiTextHeadingRule:38 -->The dekany</h2> | ||
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Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An inquiry of someone who has published on the Johnson graphs revealed he had no idea what the genus of J(5,2) was, and it may very well not be known. However, the inequalities above show the genus must be at least 1.<br /> | Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An inquiry of someone who has published on the Johnson graphs revealed he had no idea what the genus of J(5,2) was, and it may very well not be known. However, the inequalities above show the genus must be at least 1.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:40:&lt;h1&gt; --><h1 id="toc20"><a name="Eleven note scales"></a><!-- ws:end:WikiTextHeadingRule:40 -->Eleven note scales</h1> | <!-- ws:start:WikiTextHeadingRule:40:&lt;h1&gt; --><h1 id="toc20"><a name="Eleven note scales"></a><!-- ws:end:WikiTextHeadingRule:40 -->Eleven note scales</h1> | ||
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The automorphism group of order 24 is the direct product of an involution and the group of the hexagon, which act on disjoint notes of the scale. The involution is (0,1)(5,6) and the hexagon group (dihedral group of order 12) permutes the cycle (2,7,4,9,3,8); this cycle together with the two involutions (2,3),(7,9) and (3,4),(7,8) generate the hexagon group.<br /> | The automorphism group of order 24 is the direct product of an involution and the group of the hexagon, which act on disjoint notes of the scale. The involution is (0,1)(5,6) and the hexagon group (dihedral group of order 12) permutes the cycle (2,7,4,9,3,8); this cycle together with the two involutions (2,3),(7,9) and (3,4),(7,8) generate the hexagon group.<br /> | ||
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