Graph-theoretic properties of scales: Difference between revisions

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==The diatonic scale (Meantone[7])==

The diatonic scale in 31edo consists of the notes 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of ~~7edo ) in either the~~ 5 or 7 ~~limits~~. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.

The diatonic scale in 31edo consists of the notes 0, 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of (5 or 7 limit) 7edo. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.

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.

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==Star==

Star is a scale of[[Starling temperaments#Valentine temperament-11-limit|11-limit valentine temperament]], which in [[77edo|77et]] is 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.

Star is a scale of [[Starling temperaments#Valentine temperament-11-limit|11-limit valentine temperament]], which in [[77edo|77et]] 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.

[[image:star.png]]

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==The marveldene==

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 [[The Marveldene|marveldene]]. An excellent tuning for marvel is [[166edo|166et]], 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 4-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.</pre></div>

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 [[The Marveldene|marveldene]]. An excellent tuning for marvel is [[166edo|166et]], 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 4-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.

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].</pre></div>

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<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><a href="#Graph of a scale">Graph of a scale</a> | <a href="#Connectivity">Connectivity</a> | <a href="#The Characteristic Polynomial">The Characteristic Polynomial</a> | <a href="#The Laplace Spectrum">The Laplace Spectrum</a> | <a href="#The Genus">The Genus</a> | <a href="#The Automorphism Group">The Automorphism Group</a> | <a href="#Examples">Examples</a>

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<br />

<h2 id="toc8"><a name="Examples-The diatonic scale (Meantone[7])"></a>The diatonic scale (Meantone[7])</h2>

The diatonic scale in 31edo consists of the notes 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of ~~7edo ) in either the~~ 5 or 7 ~~limits~~. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.<br />

The diatonic scale in 31edo consists of the notes 0, 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of (5 or 7 limit) 7edo. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.<br />

<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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<br />

<h2 id="toc9"><a name="Examples-Star"></a>Star</h2>

Star 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 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 />

Star 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 />

<br />

<img src="/file/view/star.png/359553295/star.png" alt="star.png" title="star.png" /><br />

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<h2 id="toc11"><a name="Examples-The marveldene"></a>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 4-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.</body></html></pre></div>

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 4-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 />

<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].</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <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>2012-08-24 12:44:44 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-24 13:35:41 UTC</tt>.<br>
-: The original revision id was <tt>359713869</tt>.<br>
+: The original revision id was <tt>359725711</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 63: / Line 63: @@
 ==The diatonic scale (Meantone[7])==
-The diatonic scale in 31edo consists of the notes 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of 7edo ) in either the 5 or 7 limits. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.
+The diatonic scale in 31edo consists of the notes 0, 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of (5 or 7 limit) 7edo. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.
 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.
@@ Line 70: / Line 70: @@
 ==Star==
-Star is a scale of[[Starling temperaments#Valentine temperament-11-limit|11-limit valentine temperament]], which in [[77edo|77et]] is 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.
+Star is a scale of [[Starling temperaments#Valentine temperament-11-limit|11-limit valentine temperament]], which in [[77edo|77et]] 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.
 [[image:star.png]]
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 ==The marveldene==
-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 [[The Marveldene|marveldene]]. An excellent tuning for marvel is [[166edo|166et]], 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 4-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.</pre></div>
+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 [[The Marveldene|marveldene]]. An excellent tuning for marvel is [[166edo|166et]], 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 4-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.
+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].</pre></div>
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 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Graph-theoretic properties of scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:24:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;a href="#Graph of a scale"&gt;Graph of a scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Connectivity"&gt;Connectivity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#The Characteristic Polynomial"&gt;The Characteristic Polynomial&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#The Laplace Spectrum"&gt;The Laplace Spectrum&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#The Genus"&gt;The Genus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#The Automorphism Group"&gt;The Automorphism Group&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Examples"&gt;Examples&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Examples-The diatonic scale (Meantone[7])"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;The diatonic scale (Meantone[7])&lt;/h2&gt;
-The diatonic scale in 31edo consists of the notes 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of 7edo ) in either the 5 or 7 limits. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.&lt;br /&gt;
+The diatonic scale in 31edo consists of the notes 0, 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of (5 or 7 limit) 7edo. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.&lt;br /&gt;
 &lt;br /&gt;
 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.&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Examples-Star"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Star&lt;/h2&gt;
-Star is a scale of&lt;a class="wiki_link" href="/Starling%20temperaments#Valentine temperament-11-limit"&gt;11-limit valentine temperament&lt;/a&gt;, which in &lt;a class="wiki_link" href="/77edo"&gt;77et&lt;/a&gt; is 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.&lt;br /&gt;
+Star is a scale of &lt;a class="wiki_link" href="/Starling%20temperaments#Valentine temperament-11-limit"&gt;11-limit valentine temperament&lt;/a&gt;, which in &lt;a class="wiki_link" href="/77edo"&gt;77et&lt;/a&gt; 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.&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Examples-The marveldene"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;The marveldene&lt;/h2&gt;
-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 &lt;a class="wiki_link" href="/The%20Marveldene"&gt;marveldene&lt;/a&gt;. An excellent tuning for marvel is &lt;a class="wiki_link" href="/166edo"&gt;166et&lt;/a&gt;, 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 4-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.&lt;/body&gt;&lt;/html&gt;</pre></div>
+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 &lt;a class="wiki_link" href="/The%20Marveldene"&gt;marveldene&lt;/a&gt;. An excellent tuning for marvel is &lt;a class="wiki_link" href="/166edo"&gt;166et&lt;/a&gt;, 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 4-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.&lt;br /&gt;
+&lt;br /&gt;
+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].&lt;/body&gt;&lt;/html&gt;</pre></div>