Graph-theoretic properties of scales: Difference between revisions

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 03:~~05:58~~ UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 13:03:45 UTC</tt>.<br>

: The original revision id was <tt>~~359389101~~</tt>.<br>

: The original revision id was <tt>359476563</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 18:

A [[http://en.wikipedia.org/wiki/Clique_%28graph_theory%29|clique]] in a graph is a subgraph such that there is an edge connecting every vertex; that is, the subgraph is a complete graph. In music terms, a clique is a [[dyadic chord]]. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the [[http://en.wikipedia.org/wiki/Clique_problem|clique problem]], and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices.

Among all cliques, the [[http://mathworld.wolfram.com/MaximalClique.html|maximal cliques]], those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A //maximum clique// is a clique of largest size; these are always maximal but the converse does not always hold.

Among all cliques, the [[http://mathworld.wolfram.com/MaximalClique.html|maximal cliques]], those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A //maximum clique// is a clique of largest size; these are always maximal but the converse does not always hold. Given any set of sets, we may form a graph with these as the set of vertices, by drawing an edge between any two sets with a nonempty intersection. In this way we can create graphs of all k-cliques (k note dyadic chords), maximal k-cliques, all maximal cliques, and so forth. Such graphs can illuminate harmonic relationships.

=The Characteristic Polynomial=

Line 81:

A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_%28graph_theory%29" rel="nofollow">clique</a> in a graph is a subgraph such that there is an edge connecting every vertex; that is, the subgraph is a complete graph. In music terms, a clique is a <a class="wiki_link" href="/dyadic%20chord">dyadic chord</a>. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_problem" rel="nofollow">clique problem</a>, and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices.<br />

<br />

Among all cliques, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/MaximalClique.html" rel="nofollow">maximal cliques</a>, those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A <em>maximum clique</em> is a clique of largest size; these are always maximal but the converse does not always hold.<br />

Among all cliques, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/MaximalClique.html" rel="nofollow">maximal cliques</a>, those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A <em>maximum clique</em> is a clique of largest size; these are always maximal but the converse does not always hold. Given any set of sets, we may form a graph with these as the set of vertices, by drawing an edge between any two sets with a nonempty intersection. In this way we can create graphs of all k-cliques (k note dyadic chords), maximal k-cliques, all maximal cliques, and so forth. Such graphs can illuminate harmonic relationships.<br />

<br />

<h1 id="toc2"><a name="The Characteristic Polynomial"></a>The Characteristic Polynomial</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 03:05:58 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 13:03:45 UTC</tt>.<br>
-: The original revision id was <tt>359389101</tt>.<br>
+: The original revision id was <tt>359476563</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 18: / Line 18: @@
 A [[http://en.wikipedia.org/wiki/Clique_%28graph_theory%29|clique]] in a graph is a subgraph such that there is an edge connecting every vertex; that is, the subgraph is a complete graph. In music terms, a clique is a [[dyadic chord]]. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the [[http://en.wikipedia.org/wiki/Clique_problem|clique problem]], and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices.
-Among all cliques, the [[http://mathworld.wolfram.com/MaximalClique.html|maximal cliques]], those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A //maximum clique// is a clique of largest size; these are always maximal but the converse does not always hold.
+Among all cliques, the [[http://mathworld.wolfram.com/MaximalClique.html|maximal cliques]], those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A //maximum clique// is a clique of largest size; these are always maximal but the converse does not always hold. Given any set of sets, we may form a graph with these as the set of vertices, by drawing an edge between any two sets with a nonempty intersection. In this way we can create graphs of all k-cliques (k note dyadic chords), maximal k-cliques, all maximal cliques, and so forth. Such graphs can illuminate harmonic relationships.
 =The Characteristic Polynomial=
@@ Line 81: / Line 81: @@
 A &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_%28graph_theory%29" rel="nofollow"&gt;clique&lt;/a&gt; in a graph is a subgraph such that there is an edge connecting every vertex; that is, the subgraph is a complete graph. In music terms, a clique is a &lt;a class="wiki_link" href="/dyadic%20chord"&gt;dyadic chord&lt;/a&gt;. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_problem" rel="nofollow"&gt;clique problem&lt;/a&gt;, and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices.&lt;br /&gt;
 &lt;br /&gt;
-Among all cliques, the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/MaximalClique.html" rel="nofollow"&gt;maximal cliques&lt;/a&gt;, those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A &lt;em&gt;maximum clique&lt;/em&gt; is a clique of largest size; these are always maximal but the converse does not always hold.&lt;br /&gt;
+Among all cliques, the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/MaximalClique.html" rel="nofollow"&gt;maximal cliques&lt;/a&gt;, those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A &lt;em&gt;maximum clique&lt;/em&gt; is a clique of largest size; these are always maximal but the converse does not always hold. Given any set of sets, we may form a graph with these as the set of vertices, by drawing an edge between any two sets with a nonempty intersection. In this way we can create graphs of all k-cliques (k note dyadic chords), maximal k-cliques, all maximal cliques, and so forth. Such graphs can illuminate harmonic relationships.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="The Characteristic Polynomial"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;The Characteristic Polynomial&lt;/h1&gt;