Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 360081091 - 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:genewardsmith|genewardsmith]] and made on <tt>2012-08- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-27 13:16:38 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>360215369</tt>.<br> | ||
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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 automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5. | The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5. | ||
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. | 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 [[image:myna_89edo.jpg]]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. | ||
[[image:dekany.png]] | [[image:dekany.png]] | ||
==Myna[11]== | |||
Myna[11], the 11-note MOS of [[Starling temperaments#x5-limit (Mynic)-11-limit|myna temperament]], can be specified in the [[89edo|89et]] tuning as the notes 0, 3, 20, 23, 26, 43, 46, 63, 66, 69, 86, 89. Using the 11-limit diamond as a consonance set gives a graph with 40 edges corresponding to 40 dyads. Its connectivities are 6 ≤ 7 ≤ 7, It has eight vertices of degree 7, and three of degree 8, with a radius and diameter both 2. | |||
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. | |||
==The marveldene== | ==The marveldene== | ||
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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.</pre></div> | The graph of Orwell[13] is 10-regular, has 65 edges, with connectivities 9.058 ≤ 10 ≤ 10, and radius and diameter both 2.</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: | <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:36:&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:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 1em;"><a href="#Graph of a scale">Graph of a scale</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 1em;"><a href="#Connectivity">Connectivity</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#The Characteristic Polynomial">The Characteristic Polynomial</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#The Laplace Spectrum">The Laplace Spectrum</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 1em;"><a href="#The Genus">The Genus</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 1em;"><a href="#The Automorphism Group">The Automorphism Group</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 1em;"><a href="#Examples">Examples</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 2em;"><a href="#Examples-The Zarlino scale">The Zarlino scale</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#Examples-The diatonic scale (Meantone[7])">The diatonic scale (Meantone[7])</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Examples-Star">Star</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Examples-Oktone">Oktone</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#Examples-Orwell[9]">Orwell[9]</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 2em;"><a href="#Examples-Magic[10]">Magic[10]</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#Examples-The dekany">The dekany</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 2em;"><a href="#Examples-Myna[11]">Myna[11]</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 2em;"><a href="#Examples-The marveldene">The marveldene</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 2em;"><a href="#Examples-Magic[13]">Magic[13]</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 2em;"><a href="#Examples-Orwell[13]">Orwell[13]</a></div> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Graph of a scale"></a><!-- ws:end:WikiTextHeadingRule:0 -->Graph of a scale</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Graph of a scale"></a><!-- ws:end:WikiTextHeadingRule:0 -->Graph of a scale</h1> | ||
Given a <a class="wiki_link" href="/periodic%20scale">periodic scale</a>, meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by <a class="wiki_link" href="/Scala">Scala</a>. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that &quot;5/4&quot; represents {...5/8, 5/4, 5/2, 5, 10 ...} and both &quot;1&quot; and &quot;2&quot; mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 &lt; s &lt; 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class.<br /> | Given a <a class="wiki_link" href="/periodic%20scale">periodic scale</a>, meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by <a class="wiki_link" href="/Scala">Scala</a>. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that &quot;5/4&quot; represents {...5/8, 5/4, 5/2, 5, 10 ...} and both &quot;1&quot; and &quot;2&quot; mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 &lt; s &lt; 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class.<br /> | ||
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A <a class="wiki_link" href="/dyadic%20chord">dyadic chord</a> pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below:<br /> | A <a class="wiki_link" href="/dyadic%20chord">dyadic chord</a> pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below:<br /> | ||
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The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding. On the other hand, a tetrad is of genus 0, since it can be drawn on a sphere as the verticies of a tetrahedron.<br /> | The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding. On the other hand, a tetrad is of genus 0, since it can be drawn on a sphere as the verticies of a tetrahedron.<br /> | ||
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The Zarlino scale, or &quot;just diatonic&quot; as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.<br /> | The Zarlino scale, or &quot;just diatonic&quot; as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Examples-The diatonic scale (Meantone[7])"></a><!-- ws:end:WikiTextHeadingRule:16 -->The diatonic scale (Meantone[7])</h2> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Examples-The diatonic scale (Meantone[7])"></a><!-- ws:end:WikiTextHeadingRule:16 -->The diatonic scale (Meantone[7])</h2> | ||
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The genus of the 7-limit diatonic scale is 1, with maximal genus of 4. The connectivities go 3.198 ≤ 4 ≤ 4, and the radius and diameter are both 2.<br /> | The genus of the 7-limit diatonic scale is 1, with maximal genus of 4. The connectivities go 3.198 ≤ 4 ≤ 4, and the radius and diameter are both 2.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Examples-Star"></a><!-- ws:end:WikiTextHeadingRule:18 -->Star</h2> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Examples-Star"></a><!-- ws:end:WikiTextHeadingRule:18 -->Star</h2> | ||
<a class="wiki_link" href="/Star">Star</a> is a scale of <a class="wiki_link" href="/Starling%20temperaments#Valentine temperament-11-limit">11-limit valentine temperament</a>, which in <a class="wiki_link" href="/77edo">77et</a> is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads.<br /> | <a class="wiki_link" href="/Star">Star</a> is a scale of <a class="wiki_link" href="/Starling%20temperaments#Valentine temperament-11-limit">11-limit valentine temperament</a>, which in <a class="wiki_link" href="/77edo">77et</a> is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group [2^4]S4. Its graph is 6-regular, with every vertex connected to six others, and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected. It has 16 maximal cliques, all of which are tetrads, and the automorphism group acts faithfully on the tetrads.<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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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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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 /> | 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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<!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Examples-The dekany"></a><!-- ws:end:WikiTextHeadingRule:26 -->The dekany</h2> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Examples-The dekany"></a><!-- ws:end:WikiTextHeadingRule:26 -->The dekany</h2> | ||
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The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5.<br /> | The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5.<br /> | ||
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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 <!-- ws:start:WikiTextLocalImageRule:63:&lt;img src=&quot;/file/view/myna_89edo.jpg/271260474/myna_89edo.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/myna_89edo.jpg/271260474/myna_89edo.jpg" alt="myna_89edo.jpg" title="myna_89edo.jpg" /><!-- ws:end:WikiTextLocalImageRule:63 -->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:28:&lt;h2&gt; --><h2 id="toc14"><a name="Examples-Myna[11]"></a><!-- ws:end:WikiTextHeadingRule:28 -->Myna[11]</h2> | |||
Myna[11], the 11-note MOS of <a class="wiki_link" href="/Starling%20temperaments#x5-limit (Mynic)-11-limit">myna temperament</a>, can be specified in the <a class="wiki_link" href="/89edo">89et</a> tuning as the notes 0, 3, 20, 23, 26, 43, 46, 63, 66, 69, 86, 89. Using the 11-limit diamond as a consonance set gives a graph with 40 edges corresponding to 40 dyads. Its connectivities are 6 ≤ 7 ≤ 7, It has eight vertices of degree 7, and three of degree 8, with a radius and diameter both 2.<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 /> | |||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Examples-The marveldene"></a><!-- ws:end:WikiTextHeadingRule:30 -->The marveldene</h2> | ||
The Ellis duodene is the 12-note 5-limit scale 1-16/15-9/8-6/5-5/4-4/3-45/32-3/2-8/5-5/3-9/5-15/8-2, and marvel tempering it leads to <a class="wiki_link" href="/The%20Marveldene">marveldene</a>. An excellent tuning for marvel is <a class="wiki_link" href="/166edo">166et</a>, and in that the scale becomes 0, 16, 28, 44, 53, 69, 81, 97, 113, 122, 141, 150, 166. If we use the 15-limit consonance set, {16, 18, 19, 21, 23, 25, 28, 32, 34, 37, 40, 44, 48, 50, 53, 58, 60, 63, 69, 74, 76, 78, 81, 85, 88, 90, 92, 97, 103, 106, 108, 113, 116, 118, 122, 126, 129, 132, 134, 138, 141, 143, 145, 147, 148, 150}, we obtain a graph with 55 edges and an automorphism group of order 32. Abstractly, the automorphism group is the direct product of the group of the square with the Klein four-group; as a permutation group it is the square (dihedral group D8 of order 8) part, generated by {(1,2)(5,6)(8,11)(9,10), (1,5), (2,6)}, and the two involutions flipping two notes, (3,4) flipping Eb and E, and (7,12) flipping C and G.<br /> | The Ellis duodene is the 12-note 5-limit scale 1-16/15-9/8-6/5-5/4-4/3-45/32-3/2-8/5-5/3-9/5-15/8-2, and marvel tempering it leads to <a class="wiki_link" href="/The%20Marveldene">marveldene</a>. An excellent tuning for marvel is <a class="wiki_link" href="/166edo">166et</a>, and in that the scale becomes 0, 16, 28, 44, 53, 69, 81, 97, 113, 122, 141, 150, 166. If we use the 15-limit consonance set, {16, 18, 19, 21, 23, 25, 28, 32, 34, 37, 40, 44, 48, 50, 53, 58, 60, 63, 69, 74, 76, 78, 81, 85, 88, 90, 92, 97, 103, 106, 108, 113, 116, 118, 122, 126, 129, 132, 134, 138, 141, 143, 145, 147, 148, 150}, we obtain a graph with 55 edges and an automorphism group of order 32. Abstractly, the automorphism group is the direct product of the group of the square with the Klein four-group; as a permutation group it is the square (dihedral group D8 of order 8) part, generated by {(1,2)(5,6)(8,11)(9,10), (1,5), (2,6)}, and the two involutions flipping two notes, (3,4) flipping Eb and E, and (7,12) flipping C and G.<br /> | ||
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The connectivites of the marveldene go 6.222 ≤ 7 ≤ 8, and it has a radius of 1 and a diameter of 2, with center {0, 7}, ie C and G. The largest element of the Laplace spectrum is 12, so the complementary graph is not connected. It has eight maximum cliques, the septads [0, 1, 3, 5, 7, 8, 11], [0, 1, 3, 5, 7, 9, 11], [0, 1, 4, 5, 7, 8, 11], [0, 1, 4, 5, 7, 9, 11], [0, 2, 3, 6, 7, 8, 10], [0, 2, 3, 6, 7, 8, 11], [0, 2, 4, 6, 7, 8, 10] and [0, 2, 4, 6, 7, 8, 11], and two maximal pentads, [0, 3, 7, 9, 10] and [0, 4, 7, 9, 10].<br /> | The connectivites of the marveldene go 6.222 ≤ 7 ≤ 8, and it has a radius of 1 and a diameter of 2, with center {0, 7}, ie C and G. The largest element of the Laplace spectrum is 12, so the complementary graph is not connected. It has eight maximum cliques, the septads [0, 1, 3, 5, 7, 8, 11], [0, 1, 3, 5, 7, 9, 11], [0, 1, 4, 5, 7, 8, 11], [0, 1, 4, 5, 7, 9, 11], [0, 2, 3, 6, 7, 8, 10], [0, 2, 3, 6, 7, 8, 11], [0, 2, 4, 6, 7, 8, 10] and [0, 2, 4, 6, 7, 8, 11], and two maximal pentads, [0, 3, 7, 9, 10] and [0, 4, 7, 9, 10].<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Examples-Magic[13]"></a><!-- ws:end:WikiTextHeadingRule:32 -->Magic[13]</h2> | ||
Magic[13] is the 13-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, which in <a class="wiki_link" href="/104edo">104et</a> has notes 0, 5, 10, 28, 33, 38, 43, 61, 66, 71, 76, 94, 99, 104. Using 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} as the consonance set, we obtain a graph on 13 notes with 61 edges representing dyads. It has algebraic, vertex and edge connectivities all 8, and a disconnected graph complement. The radius and diameter are both 2.<br /> | Magic[13] is the 13-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, which in <a class="wiki_link" href="/104edo">104et</a> has notes 0, 5, 10, 28, 33, 38, 43, 61, 66, 71, 76, 94, 99, 104. Using 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} as the consonance set, we obtain a graph on 13 notes with 61 edges representing dyads. It has algebraic, vertex and edge connectivities all 8, and a disconnected graph complement. The radius and diameter are both 2.<br /> | ||
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It has a disparate collection of 36 maximal cliques ranging from triads to hexads, and an automorphism group which abstractly is the direct product of two square groups. However, as a permutation group the two cycles on which the automorphisms act as the square group, (3, 7, 6, 10) and (4, 8, 5, 9), don't appear in isolation, and the group actually moves everything but the central note of the MOS, note 0 in this labeling.<br /> | It has a disparate collection of 36 maximal cliques ranging from triads to hexads, and an automorphism group which abstractly is the direct product of two square groups. However, as a permutation group the two cycles on which the automorphisms act as the square group, (3, 7, 6, 10) and (4, 8, 5, 9), don't appear in isolation, and the group actually moves everything but the central note of the MOS, note 0 in this labeling.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Examples-Orwell[13]"></a><!-- ws:end:WikiTextHeadingRule:34 -->Orwell[13]</h2> | ||
Orwell[13], the 13-note MOS of orwell, has four more notes but the same set of 11-limit consonances as Orwell[9]. Now we have 0, 5, 7, 12, 17, 19, 24, 29, 34, 36, 41, 46, 48, 53, and while the graph of the scale still has a lot of symmetry, it is of an entirely different kind. This time the symmetry group is analogous to that of the diatonic scale, being the dihedral group D26 on 13 points; as with the 7-limit diatonic scale, the symmetries of Orwell[13] appear as inversion and the ability to transpose to any key while retaining the dyadic character of all dyadic chords, which however are much more numerous in kind and number. In particular, Orwell[13] has 26 maximal pentads and 13 maximal hexads on which the automorphism group faithfully acts.<br /> | Orwell[13], the 13-note MOS of orwell, has four more notes but the same set of 11-limit consonances as Orwell[9]. Now we have 0, 5, 7, 12, 17, 19, 24, 29, 34, 36, 41, 46, 48, 53, and while the graph of the scale still has a lot of symmetry, it is of an entirely different kind. This time the symmetry group is analogous to that of the diatonic scale, being the dihedral group D26 on 13 points; as with the 7-limit diatonic scale, the symmetries of Orwell[13] appear as inversion and the ability to transpose to any key while retaining the dyadic character of all dyadic chords, which however are much more numerous in kind and number. In particular, Orwell[13] has 26 maximal pentads and 13 maximal hexads on which the automorphism group faithfully acts.<br /> | ||
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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.</body></html></pre></div> | The graph of Orwell[13] is 10-regular, has 65 edges, with connectivities 9.058 ≤ 10 ≤ 10, and radius and diameter both 2.</body></html></pre></div> | ||