Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 402410470 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 402492712 - 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>2013-01-29 16:58:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>402492712</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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[[file:graph of zeus7tri.pdf]] | [[file:graph of zeus7tri.pdf]] | ||
//[[http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3|Benny]]// | ////[[http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3|Benny]]//// | ||
==Archchro== | ==Archchro== | ||
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[[image:http://upload.wikimedia.org/wikipedia/commons/a/a0/16-cell.gif]] | [[image:http://upload.wikimedia.org/wikipedia/commons/a/a0/16-cell.gif]] | ||
//[[http://micro.soonlabel.com/star/20120830-77et-star.mp3|77et Star]]// by [[Chris Vaisvil]] | ////[[http://micro.soonlabel.com/star/20120830-77et-star.mp3|77et Star]]//// by [[Chris Vaisvil]] | ||
==Oktone== | ==Oktone== | ||
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[[image:oktony.png]] | [[image:oktony.png]] | ||
//[[http://archive.org/download/HighOktoneElgar/oktelg.mp3|High Oktone Elgar]]// | ////[[http://archive.org/download/HighOktoneElgar/oktelg.mp3|High Oktone Elgar]]//// | ||
==Zeus8tri== | ==Zeus8tri== | ||
[[Zeus8tri]] is the tempering in zeus of [11/10, 6/5, 21/16, 16/11, 8/5, 7/4, 21/11, 2]. In 99et, that leads to scale steps of | [[Zeus8tri]] is the tempering in zeus of [11/10, 6/5, 21/16, 16/11, 8/5, 7/4, 21/11, 2]. In 99et, that leads to scale steps of 0, 13, 26, 39, 54, 67, 80, 93, 99, with the same consonance set as zeus7tri. The automorphism group is of order 48, the direct product of an involution (0,7)(3,4) and the symmetric group on the four elements in the center (in the sense of graph theory) of the graph, {1, 2, 5, 6}. It has connectivities 4.586 ≤ 5 ≤ 5, a radius of 1 and a diameter of 2. | ||
Zeus8tri has three maximal cliques, the hexads [0, 1, 2, 4, 5, 6], [1, 2, 3, 4, 5, 6], [1, 2, 3, 5, 6, 7], each of which consists of the four notes in the center plus {0, 4}, {3, 4}, or {3, 7}. Instead of having a single circle of thirds like zeus7tri, it has two, both of which are the 15-limit valinorsmic tetrads with steps 6/5, 11/9, 6/5, 8/7 which we have not counted among the (11-limit) chords of zeus8tri. Adding 126/125 to the commas of zeus leads to valentine temperament, which does not damage the tuning accuracy of zeus very much. In valentine, zeus8tri becomes a scale of Graham complexity 13, with generator steps -7, -5, -3, -1, 0, 2, 4, 6. | Zeus8tri has three maximal cliques, the hexads [0, 1, 2, 4, 5, 6], [1, 2, 3, 4, 5, 6], [1, 2, 3, 5, 6, 7], each of which consists of the four notes in the center plus {0, 4}, {3, 4}, or {3, 7}. Instead of having a single circle of thirds like zeus7tri, it has two, both of which are the 15-limit valinorsmic tetrads with steps 6/5, 11/9, 6/5, 8/7 which we have not counted among the (11-limit) chords of zeus8tri. Adding 126/125 to the commas of zeus leads to valentine temperament, which does not damage the tuning accuracy of zeus very much. In valentine, zeus8tri becomes a scale of Graham complexity 13, with generator steps -7, -5, -3, -1, 0, 2, 4, 6. | ||
[[image:graph of zeus8tri.png width="444" height="628"]] | |||
[[file:graph of zeus8tri.pdf]] | |||
=Nine note scales= | =Nine note scales= | ||
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[[file:graph of orwell-9 color-coded.pdf]] | [[file:graph of orwell-9 color-coded.pdf]] | ||
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Grood/Mountain_Village_TarkanGrood.mp3|Mountain Village]]// by [[Tarkan Grood]] | ////[[http://micro.soonlabel.com/gene_ward_smith/Others/Grood/Mountain_Village_TarkanGrood.mp3|Mountain Village]]//// by [[Tarkan Grood]] | ||
//[[http://micro.soonlabel.com/gene_ward_smith/transformers/swing-orwell9.mp3|Swing in Orwell-9]]// | ////[[http://micro.soonlabel.com/gene_ward_smith/transformers/swing-orwell9.mp3|Swing in Orwell-9]]//// | ||
=Ten note scales= | =Ten note scales= | ||
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The graph of Orwell[13] is 10-regular, has 65 edges, with connectivities 9.058 ≤ 10 ≤ 10, and radius and diameter both 2. | The graph of Orwell[13] is 10-regular, has 65 edges, with connectivities 9.058 ≤ 10 ≤ 10, and radius and diameter both 2. | ||
//[[http://archive.org/download/TunicataAndFugue/TunicataAndFugueVer2.mp3|Tunicata and Fugue]]// by [[Peter Kosmorsky]] | ////[[http://archive.org/download/TunicataAndFugue/TunicataAndFugueVer2.mp3|Tunicata and Fugue]]//// by [[Peter Kosmorsky]] | ||
//[[http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3|Orwellian Cameras]]// by [[Chris Vaisvil]]</pre></div> | ////[[http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3|Orwellian Cameras]]//// by [[Chris Vaisvil]]</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Graph-theoretic properties of scales</title></head><body><!-- ws:start:WikiTextTocRule:58:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 1em;"><a href="#Graph of a scale">Graph of a scale</a></div> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Graph-theoretic properties of scales</title></head><body><!-- ws:start:WikiTextTocRule:58:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 1em;"><a href="#Graph of a scale">Graph of a scale</a></div> | ||
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<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3" rel="nofollow">Benny</a><br /> | |||
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<a class="wiki_link_ext" href="http://micro.soonlabel.com/star/20120830-77et-star.mp3" rel="nofollow">77et Star</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> | |||
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<!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Eight note scales-Oktone"></a><!-- ws:end:WikiTextHeadingRule:28 -->Oktone</h2> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Eight note scales-Oktone"></a><!-- ws:end:WikiTextHeadingRule:28 -->Oktone</h2> | ||
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<a class="wiki_link_ext" href="http://archive.org/download/HighOktoneElgar/oktelg.mp3" rel="nofollow">High Oktone Elgar</a><br /> | |||
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<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Eight note scales-Zeus8tri"></a><!-- ws:end:WikiTextHeadingRule:30 -->Zeus8tri</h2> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Eight note scales-Zeus8tri"></a><!-- ws:end:WikiTextHeadingRule:30 -->Zeus8tri</h2> | ||
<a class="wiki_link" href="/Zeus8tri">Zeus8tri</a> is the tempering in zeus of [11/10, 6/5, 21/16, 16/11, 8/5, 7/4, 21/11, 2]. In 99et, that leads to scale steps of | <a class="wiki_link" href="/Zeus8tri">Zeus8tri</a> is the tempering in zeus of [11/10, 6/5, 21/16, 16/11, 8/5, 7/4, 21/11, 2]. In 99et, that leads to scale steps of 0, 13, 26, 39, 54, 67, 80, 93, 99, with the same consonance set as zeus7tri. The automorphism group is of order 48, the direct product of an involution (0,7)(3,4) and the symmetric group on the four elements in the center (in the sense of graph theory) of the graph, {1, 2, 5, 6}. It has connectivities 4.586 ≤ 5 ≤ 5, a radius of 1 and a diameter of 2.<br /> | ||
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Zeus8tri has three maximal cliques, the hexads [0, 1, 2, 4, 5, 6], [1, 2, 3, 4, 5, 6], [1, 2, 3, 5, 6, 7], each of which consists of the four notes in the center plus {0, 4}, {3, 4}, or {3, 7}. Instead of having a single circle of thirds like zeus7tri, it has two, both of which are the 15-limit valinorsmic tetrads with steps 6/5, 11/9, 6/5, 8/7 which we have not counted among the (11-limit) chords of zeus8tri. Adding 126/125 to the commas of zeus leads to valentine temperament, which does not damage the tuning accuracy of zeus very much. In valentine, zeus8tri becomes a scale of Graham complexity 13, with generator steps -7, -5, -3, -1, 0, 2, 4, 6.<br /> | Zeus8tri has three maximal cliques, the hexads [0, 1, 2, 4, 5, 6], [1, 2, 3, 4, 5, 6], [1, 2, 3, 5, 6, 7], each of which consists of the four notes in the center plus {0, 4}, {3, 4}, or {3, 7}. Instead of having a single circle of thirds like zeus7tri, it has two, both of which are the 15-limit valinorsmic tetrads with steps 6/5, 11/9, 6/5, 8/7 which we have not counted among the (11-limit) chords of zeus8tri. Adding 126/125 to the commas of zeus leads to valentine temperament, which does not damage the tuning accuracy of zeus very much. In valentine, zeus8tri becomes a scale of Graham complexity 13, with generator steps -7, -5, -3, -1, 0, 2, 4, 6.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Nine note scales"></a><!-- ws:end:WikiTextHeadingRule:32 -->Nine note scales</h1> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Nine note scales"></a><!-- ws:end:WikiTextHeadingRule:32 -->Nine note scales</h1> | ||
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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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<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Grood/Mountain_Village_TarkanGrood.mp3" rel="nofollow">Mountain Village</a> by <a class="wiki_link" href="/Tarkan%20Grood">Tarkan Grood</a><br /> | |||
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/transformers/swing-orwell9.mp3" rel="nofollow">Swing in Orwell-9</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:40:&lt;h2&gt; --><h2 id="toc20"><a name="Ten note scales-The dekany"></a><!-- ws:end:WikiTextHeadingRule:40 -->The dekany</h2> | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Ten note scales-The dekany"></a><!-- ws:end:WikiTextHeadingRule:40 -->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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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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<!-- ws:start:WikiTextHeadingRule:46:&lt;h1&gt; --><h1 id="toc23"><a name="Twelve note scales"></a><!-- ws:end:WikiTextHeadingRule:46 -->Twelve note scales</h1> | <!-- ws:start:WikiTextHeadingRule:46:&lt;h1&gt; --><h1 id="toc23"><a name="Twelve note scales"></a><!-- ws:end:WikiTextHeadingRule:46 -->Twelve note scales</h1> | ||
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The graph of Orwell[13] is 10-regular, has 65 edges, with connectivities 9.058 ≤ 10 ≤ 10, and radius and diameter both 2.<br /> | The graph of Orwell[13] is 10-regular, has 65 edges, with connectivities 9.058 ≤ 10 ≤ 10, and radius and diameter both 2.<br /> | ||
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<a class="wiki_link_ext" href="http://archive.org/download/TunicataAndFugue/TunicataAndFugueVer2.mp3" rel="nofollow">Tunicata and Fugue</a> by <a class="wiki_link" href="/Peter%20Kosmorsky">Peter Kosmorsky</a><br /> | |||
<a class="wiki_link_ext" href="http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3" rel="nofollow">Orwellian Cameras</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></body></html></pre></div> | |||