Graph-theoretic properties of scales: Difference between revisions

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 02:29:15 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 03:05:58 UTC</tt>.<br>

: The original revision id was <tt>~~359385787~~</tt>.<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.

If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for [[http://en.wikipedia.org/wiki/Dense_graph|graph density]], 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V>2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) ~~whih~~ has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2

If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for [[http://en.wikipedia.org/wiki/Dense_graph|graph density]], 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V>2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) which has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2

=The Automorphism Group=

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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 />

<br />

If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dense_graph" rel="nofollow">graph density</a>, 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V&gt;2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) ~~whih~~ has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2<br />

If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dense_graph" rel="nofollow">graph density</a>, 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V&gt;2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) which has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2<br />

<br />

<h1 id="toc5"><a name="The Automorphism Group"></a>The Automorphism Group</h1>

@@ 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>2012-08-23 02:29:15 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 03:05:58 UTC</tt>.<br>
-: The original revision id was <tt>359385787</tt>.<br>
+: The original revision id was <tt>359389101</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 49: / Line 49: @@
 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.
-If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for [[http://en.wikipedia.org/wiki/Dense_graph|graph density]], 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V&gt;2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) whih has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2
+If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for [[http://en.wikipedia.org/wiki/Dense_graph|graph density]], 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V&gt;2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) which has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2
 =The Automorphism Group=
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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.&lt;br /&gt;
 &lt;br /&gt;
-If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dense_graph" rel="nofollow"&gt;graph density&lt;/a&gt;, 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V&amp;gt;2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) whih has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2&lt;br /&gt;
+If V is the number of notes (vertices) of the scale, and E the number of dyads (edges), then the maximum value for E is C(V, 2) = V(V - 1)/2. This suggests the definition for &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dense_graph" rel="nofollow"&gt;graph density&lt;/a&gt;, 2E/(V(V-1)). Since the maximum number of edges for a genus 0 graph with V&amp;gt;2 is 3V - 6, the maximum density for a genus 0 graph is 6(V-2)/(V(V-1)) which has series expansion 6/V - 6/V^2 - 6/V^3 ...; if the density is 6/V or greater the graph must be of a higher genus, hence, higher genus scales are to be expected in music. If each note is harmonically related to at least three other notes, an obviously desirable property which in graph-theoretic language means the minimum degree is greater than two, then the genus g ≥ E/6 - V/2 + 1. On the other hand, for a connected graph we have g ≤ (E - V + 1)/2&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="The Automorphism Group"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;The Automorphism Group&lt;/h1&gt;