Graph-theoretic properties of scales: Difference between revisions

Revision as of 14:44, 17 August 2012

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[[toc|flat]]

=Graph of a scale=
Given a [[periodic scale]], 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 [[Scala]]. 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 "5/4" represents {...5/8, 5/4, 5/2, 5, 10 ...} and both "1" and "2" 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 < s < 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. 

We now may define the **graph of the scale**, which more precisely is the graph G = G(S, C) of the scale S together with the consonance set C. This means G is a (simple) [[http://en.wikipedia.org/wiki/Graph_(mathematics)|graph]] in the sense of [[http://en.wikipedia.org/wiki/Graph_theory|graph theory]]. The vertices of G, V(G), is the set S of pitch classes, and an edge is drawn between two distinct pitch classes r and s if X⋂C ≠ ∅, where X = {x/y|x∊r, y∊s}. This means that there is a relation of consonance, as defined by C, between the pitch classes r and s. It should be noted that we have defined things assuming pitches are given multiplicatively, but we can equally well express them in logarithmic terms as for instance by cents or by steps of an [[EDO]].

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<html><head><title>Graph-theoretic properties of scales</title></head><body><!-- ws:start:WikiTextTocRule:2:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:2 --><!-- ws:start:WikiTextTocRule:3: --><a href="#Graph of a scale">Graph of a scale</a><!-- ws:end:WikiTextTocRule:3 --><!-- ws:start:WikiTextTocRule:4: -->
<!-- ws:end:WikiTextTocRule:4 --><br />
<!-- 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 />
<br />
We now may define the <strong>graph of the scale</strong>, which more precisely is the graph G = G(S, C) of the scale S together with the consonance set C. This means G is a (simple) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_(mathematics)" rel="nofollow">graph</a> in the sense of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_theory" rel="nofollow">graph theory</a>. The vertices of G, V(G), is the set S of pitch classes, and an edge is drawn between two distinct pitch classes r and s if X⋂C ≠ ∅, where X = {x/y|x∊r, y∊s}. This means that there is a relation of consonance, as defined by C, between the pitch classes r and s. It should be noted that we have defined things assuming pitches are given multiplicatively, but we can equally well express them in logarithmic terms as for instance by cents or by steps of an <a class="wiki_link" href="/EDO">EDO</a>.</body></html>

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 <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>2012-08-17 14:21:27 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-17 14:44:13 UTC</tt>.<br>
-: The original revision id was <tt>358382430</tt>.<br>
+: The original revision id was <tt>358386464</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 11: / Line 11: @@
 Given a [[periodic scale]], 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 [[Scala]]. 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 "5/4" represents {...5/8, 5/4, 5/2, 5, 10 ...} and both "1" and "2" 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.
-We now may define the **graph of the scale**, which more precisely is the graph G = G(S, C) of the scale S together with the consonance set C. This means G is a (simple) [[http://en.wikipedia.org/wiki/Graph_(mathematics)|graph]] in the sense of [[http://en.wikipedia.org/wiki/Graph_theory|graph theory]]. The vertices of G, V(G), is the set S of pitch classes, and an edge is drawn between two distinct pitch classes r and s if the pitch class X⋂C ≠ ∅, where X = {x/y|x∊r, y∊s}. This means that there is a relation of consonance, as defined by C, between the pitch classes r and s.</pre></div>
+We now may define the **graph of the scale**, which more precisely is the graph G = G(S, C) of the scale S together with the consonance set C. This means G is a (simple) [[http://en.wikipedia.org/wiki/Graph_(mathematics)|graph]] in the sense of [[http://en.wikipedia.org/wiki/Graph_theory|graph theory]]. The vertices of G, V(G), is the set S of pitch classes, and an edge is drawn between two distinct pitch classes r and s if X⋂C ≠ ∅, where X = {x/y|x∊r, y∊s}. This means that there is a relation of consonance, as defined by C, between the pitch classes r and s. It should be noted that we have defined things assuming pitches are given multiplicatively, but we can equally well express them in logarithmic terms as for instance by cents or by steps of an [[EDO]].</pre></div>
 <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">&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:2:&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:2 --&gt;&lt;!-- ws:start:WikiTextTocRule:3: --&gt;&lt;a href="#Graph of a scale"&gt;Graph of a scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:3 --&gt;&lt;!-- ws:start:WikiTextTocRule:4: --&gt;
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 Given a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/Scala"&gt;Scala&lt;/a&gt;. 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 &amp;quot;5/4&amp;quot; represents {...5/8, 5/4, 5/2, 5, 10 ...} and both &amp;quot;1&amp;quot; and &amp;quot;2&amp;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 &amp;lt; s &amp;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. &lt;br /&gt;
 &lt;br /&gt;
-We now may define the &lt;strong&gt;graph of the scale&lt;/strong&gt;, which more precisely is the graph G = G(S, C) of the scale S together with the consonance set C. This means G is a (simple) &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_(mathematics)" rel="nofollow"&gt;graph&lt;/a&gt; in the sense of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_theory" rel="nofollow"&gt;graph theory&lt;/a&gt;. The vertices of G, V(G), is the set S of pitch classes, and an edge is drawn between two distinct pitch classes r and s if the pitch class X⋂C ≠ ∅, where X = {x/y|x∊r, y∊s}. This means that there is a relation of consonance, as defined by C, between the pitch classes r and s.&lt;/body&gt;&lt;/html&gt;</pre></div>
+We now may define the &lt;strong&gt;graph of the scale&lt;/strong&gt;, which more precisely is the graph G = G(S, C) of the scale S together with the consonance set C. This means G is a (simple) &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_(mathematics)" rel="nofollow"&gt;graph&lt;/a&gt; in the sense of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_theory" rel="nofollow"&gt;graph theory&lt;/a&gt;. The vertices of G, V(G), is the set S of pitch classes, and an edge is drawn between two distinct pitch classes r and s if X⋂C ≠ ∅, where X = {x/y|x∊r, y∊s}. This means that there is a relation of consonance, as defined by C, between the pitch classes r and s. It should be noted that we have defined things assuming pitches are given multiplicatively, but we can equally well express them in logarithmic terms as for instance by cents or by steps of an &lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>

Graph-theoretic properties of scales: Difference between revisions

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