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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>2012-08-17 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-17 21:37:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>358436073</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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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. | 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]].</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]]. | ||
=Connectivity= | |||
We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be //connected// if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity λ(G) if it is possible to disconnect the graph by removing λ(G) edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity κ(G) if it is possible to disconnect the graph by removing κ(G) vertices(notes of the scale) but no smaller number will do. The graph is //k-edge-connected// if λ(G) ≥ k, and //k-vertex-connected// if κ(G) ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale.</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:4:&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:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Graph of a scale">Graph of a scale</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: --> | <a href="#Connectivity">Connectivity</a><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:7 --><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> | <!-- 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 /> | ||
<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></pre></div> | 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>.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Connectivity"></a><!-- ws:end:WikiTextHeadingRule:2 -->Connectivity</h1> | |||
We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be <em>connected</em> if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity λ(G) if it is possible to disconnect the graph by removing λ(G) edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity κ(G) if it is possible to disconnect the graph by removing κ(G) vertices(notes of the scale) but no smaller number will do. The graph is <em>k-edge-connected</em> if λ(G) ≥ k, and <em>k-vertex-connected</em> if κ(G) ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale.</body></html></pre></div> | |||
Revision as of 21:37, 17 August 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-08-17 21:37:10 UTC.
- The original revision id was 358436073.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[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]].
=Connectivity=
We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be //connected// if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity λ(G) if it is possible to disconnect the graph by removing λ(G) edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity κ(G) if it is possible to disconnect the graph by removing κ(G) vertices(notes of the scale) but no smaller number will do. The graph is //k-edge-connected// if λ(G) ≥ k, and //k-vertex-connected// if κ(G) ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale.Original HTML content:
<html><head><title>Graph-theoretic properties of scales</title></head><body><!-- ws:start:WikiTextTocRule:4:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Graph of a scale">Graph of a scale</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: --> | <a href="#Connectivity">Connectivity</a><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: -->
<!-- ws:end:WikiTextTocRule:7 --><br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><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 "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. <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>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Connectivity"></a><!-- ws:end:WikiTextHeadingRule:2 -->Connectivity</h1>
We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be <em>connected</em> if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity λ(G) if it is possible to disconnect the graph by removing λ(G) edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity κ(G) if it is possible to disconnect the graph by removing κ(G) vertices(notes of the scale) but no smaller number will do. The graph is <em>k-edge-connected</em> if λ(G) ≥ k, and <em>k-vertex-connected</em> if κ(G) ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale.</body></html>