Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 402066972 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 402148218 - 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 16:41:32 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>402148218</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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[[image:diatonic7.png]] | [[image:diatonic7.png]] | ||
==Gypsy== | ==Gypsy== | ||
[[Gypsy]] is the tempering in 7-limit marvel of the JI scale 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2. This scale is mavchrome1, the first 25/24&135/128 Fokker block, and is also the 5-limit, 7-note JI hobbit, or "jobbit". It is also Helmholtz's Chromatic and the Slovakian gypsy major. Another mode is the Slovakian gypsy minor, noted by Tartini, and still another mode is the tempering of a 7-limit scale due to Dave Keenan, keenanjust. Last but hardly least, it is the tempering of the 7-limit scale listed in the Scala catalog as "al-farabi_chrom2", and is derived from a permutation of Al Farabi's chromatic tetrachord 7/6-15/14-16/15. This kind of tetrachordal permutation was a part of the medieval Islamic theory. | [[Gypsy]] is the tempering in 7-limit marvel of the JI scale 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2. This scale is mavchrome1, the first 25/24&135/128 Fokker block, and is also the 5-limit, 7-note JI hobbit, or "jobbit". It is also Helmholtz's Chromatic and the Slovakian gypsy major. Another mode is the Slovakian gypsy minor, noted by Tartini, and still another mode is the tempering of a 7-limit scale due to Dave Keenan, keenanjust. Last but hardly least, it is the tempering of the 7-limit scale listed in the Scala catalog as "al-farabi_chrom2", and is derived from a permutation of Al Farabi's chromatic tetrachord 7/6-15/14-16/15. This kind of tetrachordal permutation was a part of the medieval Islamic theory. | ||
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The 9-limit graph has eight maximal cliques, four triads, each containing 0, and four tetrads, each containing both 1 and 6. The triads consist of 0 plus any of the dyads {2, 4}, {2, 5}, {3, 4} or {3, 5}, and the tetrads consist of {1, 6} plus any of the same set of dyads. The genus of both the 7-limit and the 9-limit graphs is 1. | The 9-limit graph has eight maximal cliques, four triads, each containing 0, and four tetrads, each containing both 1 and 6. The triads consist of 0 plus any of the dyads {2, 4}, {2, 5}, {3, 4} or {3, 5}, and the tetrads consist of {1, 6} plus any of the same set of dyads. The genus of both the 7-limit and the 9-limit graphs is 1. | ||
==Zeus7tri== | ==Zeus7tri== | ||
[[Zeus7tri]] is the tempering in zeus of 11/10, 5/4, 11/8, 3/2, 12/7, 15/8, 2. In 99et it has steps 0, 13, 32, 45, 58, 77, 90, 99, with an 11-limit consonance set {13, 13, 15, 17, 19, 22, 26, 28, 32, 35, 36, 41, 45, 48, 51, 54, 58, 63, 64, 67, 71, 73, 77, 80, 82, 84, 86, 86}. The scale contains two maximal hexads, consisting of the pentad of the notes 1 through 5 plus either 0 or 6. The automorphism group is of order 240, consisting of the direct product of the group of order two exchanging 0 and 6, and the symmetric group on the five other notes 1, 2, 3, 4, 5. It has vertex, edge and algebraic connectivities all 5. | [[Zeus7tri]] is the tempering in zeus of 11/10, 5/4, 11/8, 3/2, 12/7, 15/8, 2. In 99et it has steps 0, 13, 32, 45, 58, 77, 90, 99, with an 11-limit consonance set {13, 13, 15, 17, 19, 22, 26, 28, 32, 35, 36, 41, 45, 48, 51, 54, 58, 63, 64, 67, 71, 73, 77, 80, 82, 84, 86, 86}. The scale contains two maximal hexads, consisting of the pentad of the notes 1 through 5 plus either 0 or 6. The automorphism group is of order 240, consisting of the direct product of the group of order two exchanging 0 and 6, and the symmetric group on the five other notes 1, 2, 3, 4, 5. It has vertex, edge and algebraic connectivities all 5. | ||
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. | ||
[[http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3|Benny]] | //[[http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3|Benny]]// | ||
==Archchro== | ==Archchro== | ||
[[Archchro]] is the tempering in hemif, a 2.3.7.11.13 subgroup scale, of [[Archytas' Chromatic]]. In the 58et tuning, it has steps of 0, 3, 10, 24, 34, 37, 44, 58, with a consonance set {6, 7, 10, 11, 13, 14, 17, 20, 21, 24, 27, 31, 34, 37, 38, 41, 44, 45, 47, 48, 51, 52}. Archchro is intransitive, with one orbit consisting of notes 2, 3, and 6, and the other of notes 0, 1, 4 and 5. The three note orbit is permuted by the group of the triangle, which is to say, the symmetric group S3, and the four note orbit by the group of the square. Together these give an automorphism group of order 6 X 8 = 48. The eight permutations fixing 2, 3, and 7 are {0123456, 0123546, 1023456, 1023546, 4523016, 4523106, 5423016, 5423106}, and the six fixing 0, 1, 4 and 5 are {0123456, 0126453, 0132456, 0136452, 0162453, 0163452}. | [[Archchro]] is the tempering in hemif, a 2.3.7.11.13 subgroup scale, of [[Archytas' Chromatic]]. In the 58et tuning, it has steps of 0, 3, 10, 24, 34, 37, 44, 58, with a consonance set {6, 7, 10, 11, 13, 14, 17, 20, 21, 24, 27, 31, 34, 37, 38, 41, 44, 45, 47, 48, 51, 52}. Archchro is intransitive, with one orbit consisting of notes 2, 3, and 6, and the other of notes 0, 1, 4 and 5. The three note orbit is permuted by the group of the triangle, which is to say, the symmetric group S3, and the four note orbit by the group of the square. Together these give an automorphism group of order 6 X 8 = 48. The eight permutations fixing 2, 3, and 7 are {0123456, 0123546, 1023456, 1023546, 4523016, 4523106, 5423016, 5423106}, and the six fixing 0, 1, 4 and 5 are {0123456, 0126453, 0132456, 0136452, 0162453, 0163452}. | ||
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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]]// | ||
=Nine note scales= | =Nine note scales= | ||
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[[image:orwell9.png]] | [[image:orwell9.png]] | ||
[[image:graph of orwell-9 color-coded.png width="445" height="629"]] | |||
[[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:56:&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:56 --><!-- ws:start:WikiTextTocRule:57: --><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:56:&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:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 1em;"><a href="#Graph of a scale">Graph of a scale</a></div> | ||
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Seven note scales-Gypsy"></a><!-- ws:end:WikiTextHeadingRule:18 -->Gypsy</h2> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Seven note scales-Gypsy"></a><!-- ws:end:WikiTextHeadingRule:18 -->Gypsy</h2> | ||
<a class="wiki_link" href="/Gypsy">Gypsy</a> is the tempering in 7-limit marvel of the JI scale 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2. This scale is mavchrome1, the first 25/24&amp;135/128 Fokker block, and is also the 5-limit, 7-note JI hobbit, or &quot;jobbit&quot;. It is also Helmholtz's Chromatic and the Slovakian gypsy major. Another mode is the Slovakian gypsy minor, noted by Tartini, and still another mode is the tempering of a 7-limit scale due to Dave Keenan, keenanjust. Last but hardly least, it is the tempering of the 7-limit scale listed in the Scala catalog as &quot;al-farabi_chrom2&quot;, and is derived from a permutation of Al Farabi's chromatic tetrachord 7/6-15/14-16/15. This kind of tetrachordal permutation was a part of the medieval Islamic theory.<br /> | <a class="wiki_link" href="/Gypsy">Gypsy</a> is the tempering in 7-limit marvel of the JI scale 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2. This scale is mavchrome1, the first 25/24&amp;135/128 Fokker block, and is also the 5-limit, 7-note JI hobbit, or &quot;jobbit&quot;. It is also Helmholtz's Chromatic and the Slovakian gypsy major. Another mode is the Slovakian gypsy minor, noted by Tartini, and still another mode is the tempering of a 7-limit scale due to Dave Keenan, keenanjust. Last but hardly least, it is the tempering of the 7-limit scale listed in the Scala catalog as &quot;al-farabi_chrom2&quot;, and is derived from a permutation of Al Farabi's chromatic tetrachord 7/6-15/14-16/15. This kind of tetrachordal permutation was a part of the medieval Islamic theory.<br /> | ||
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However arrived at, the scale in <a class="wiki_link" href="/197edo">197et</a> is 0, 19, 63, 82, 115, 134, 178, 197. It has two graphs of interest, since the graphs of 7-limit relations and of 9-limit relations are not isomorphic, but the automorphism groups (of order 16) of these graphs are. The 7-limit consonance set is {38, 44, 52, 63, 82, 96, 101, 115, 134, 145, 153, 159, 197} and the 9-limit set is {30, 33, 38, 44, 52, 63, 71, 82, 96, 101, 115, 126, 134, 145, 153, 159, 164, 167, 197}. The difference is {30, 33, 71, 126, 164, 167}. In Gypsy, the 9-limit intervals occur between 5/4 and 8/5, tempered to 9/7, and between 4/3 and 3/2.<br /> | However arrived at, the scale in <a class="wiki_link" href="/197edo">197et</a> is 0, 19, 63, 82, 115, 134, 178, 197. It has two graphs of interest, since the graphs of 7-limit relations and of 9-limit relations are not isomorphic, but the automorphism groups (of order 16) of these graphs are. The 7-limit consonance set is {38, 44, 52, 63, 82, 96, 101, 115, 134, 145, 153, 159, 197} and the 9-limit set is {30, 33, 38, 44, 52, 63, 71, 82, 96, 101, 115, 126, 134, 145, 153, 159, 164, 167, 197}. The difference is {30, 33, 71, 126, 164, 167}. In Gypsy, the 9-limit intervals occur between 5/4 and 8/5, tempered to 9/7, and between 4/3 and 3/2.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Seven note scales-Zeus7tri"></a><!-- ws:end:WikiTextHeadingRule:20 -->Zeus7tri</h2> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Seven note scales-Zeus7tri"></a><!-- ws:end:WikiTextHeadingRule:20 -->Zeus7tri</h2> | ||
<a class="wiki_link" href="/Zeus7tri">Zeus7tri</a> is the tempering in zeus of 11/10, 5/4, 11/8, 3/2, 12/7, 15/8, 2. In 99et it has steps 0, 13, 32, 45, 58, 77, 90, 99, with an 11-limit consonance set {13, 13, 15, 17, 19, 22, 26, 28, 32, 35, 36, 41, 45, 48, 51, 54, 58, 63, 64, 67, 71, 73, 77, 80, 82, 84, 86, 86}. The scale contains two maximal hexads, consisting of the pentad of the notes 1 through 5 plus either 0 or 6. The automorphism group is of order 240, consisting of the direct product of the group of order two exchanging 0 and 6, and the symmetric group on the five other notes 1, 2, 3, 4, 5. It has vertex, edge and algebraic connectivities all 5.<br /> | <a class="wiki_link" href="/Zeus7tri">Zeus7tri</a> is the tempering in zeus of 11/10, 5/4, 11/8, 3/2, 12/7, 15/8, 2. In 99et it has steps 0, 13, 32, 45, 58, 77, 90, 99, with an 11-limit consonance set {13, 13, 15, 17, 19, 22, 26, 28, 32, 35, 36, 41, 45, 48, 51, 54, 58, 63, 64, 67, 71, 73, 77, 80, 82, 84, 86, 86}. The scale contains two maximal hexads, consisting of the pentad of the notes 1 through 5 plus either 0 or 6. The automorphism group is of order 240, consisting of the direct product of the group of order two exchanging 0 and 6, and the symmetric group on the five other notes 1, 2, 3, 4, 5. It has vertex, edge and algebraic connectivities all 5.<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.<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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<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3" rel="nofollow">Benny</a><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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<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Seven note scales-Archchro"></a><!-- ws:end:WikiTextHeadingRule:22 -->Archchro</h2> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Seven note scales-Archchro"></a><!-- ws:end:WikiTextHeadingRule:22 -->Archchro</h2> | ||
<a class="wiki_link" href="/Archchro">Archchro</a> is the tempering in hemif, a 2.3.7.11.13 subgroup scale, of <a class="wiki_link" href="/Archytas%27%20Chromatic">Archytas' Chromatic</a>. In the 58et tuning, it has steps of 0, 3, 10, 24, 34, 37, 44, 58, with a consonance set {6, 7, 10, 11, 13, 14, 17, 20, 21, 24, 27, 31, 34, 37, 38, 41, 44, 45, 47, 48, 51, 52}. Archchro is intransitive, with one orbit consisting of notes 2, 3, and 6, and the other of notes 0, 1, 4 and 5. The three note orbit is permuted by the group of the triangle, which is to say, the symmetric group S3, and the four note orbit by the group of the square. Together these give an automorphism group of order 6 X 8 = 48. The eight permutations fixing 2, 3, and 7 are {0123456, 0123546, 1023456, 1023546, 4523016, 4523106, 5423016, 5423106}, and the six fixing 0, 1, 4 and 5 are {0123456, 0126453, 0132456, 0136452, 0162453, 0163452}.<br /> | <a class="wiki_link" href="/Archchro">Archchro</a> is the tempering in hemif, a 2.3.7.11.13 subgroup scale, of <a class="wiki_link" href="/Archytas%27%20Chromatic">Archytas' Chromatic</a>. In the 58et tuning, it has steps of 0, 3, 10, 24, 34, 37, 44, 58, with a consonance set {6, 7, 10, 11, 13, 14, 17, 20, 21, 24, 27, 31, 34, 37, 38, 41, 44, 45, 47, 48, 51, 52}. Archchro is intransitive, with one orbit consisting of notes 2, 3, and 6, and the other of notes 0, 1, 4 and 5. The three note orbit is permuted by the group of the triangle, which is to say, the symmetric group S3, and the four note orbit by the group of the square. Together these give an automorphism group of order 6 X 8 = 48. The eight permutations fixing 2, 3, and 7 are {0123456, 0123546, 1023456, 1023546, 4523016, 4523106, 5423016, 5423106}, and the six fixing 0, 1, 4 and 5 are {0123456, 0126453, 0132456, 0136452, 0162453, 0163452}.<br /> | ||
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Archchro has four maximal cliques, the pentads 02346, 02356, 12346, and 12356. Its graph has edge, vertex and algebraic connectivity all 5, a radius of 1 and a diameter of 2, and is of genus 1.<br /> | Archchro has four maximal cliques, the pentads 02346, 02356, 12346, and 12356. Its graph has edge, vertex and algebraic connectivity all 5, a radius of 1 and a diameter of 2, and is of genus 1.<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 /> | [[<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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<a class="wiki_link_ext" href="http://archive.org/download/HighOktoneElgar/oktelg.mp3" rel="nofollow">High Oktone Elgar</a><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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<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 /> | <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 /> | ||
<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 /> | <em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/transformers/swing-orwell9.mp3" rel="nofollow">Swing in Orwell-9</a></em><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Ten note scales"></a><!-- ws:end:WikiTextHeadingRule:34 -->Ten note scales</h1> | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Ten note scales"></a><!-- ws:end:WikiTextHeadingRule:34 -->Ten note scales</h1> | ||
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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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<!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Twelve note scales"></a><!-- ws:end:WikiTextHeadingRule:44 -->Twelve note scales</h1> | <!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Twelve note scales"></a><!-- ws:end:WikiTextHeadingRule:44 -->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 /> | <em><a class="wiki_link_ext" href="http://archive.org/download/TunicataAndFugue/TunicataAndFugueVer2.mp3" rel="nofollow">Tunicata and Fugue</a></em> by <a class="wiki_link" href="/Peter%20Kosmorsky">Peter Kosmorsky</a><br /> | ||
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