Graph-theoretic properties of scales: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:~~keenanpepper~~|~~keenanpepper~~]] and made on <tt>2012-09-21 12:44:04 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-21 15:17:03 UTC</tt>.

: The original revision id was <tt>~~366711130~~</tt>.

: The original revision id was <tt>366763582</tt>.

: The revision comment was: <tt></tt>

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Drawing (embedding) the graph on a surface with no crossings would give a nice visual picture of the harmonic relations in a scale. Moreover, such an embedding can be converted into a planar diagram by allowing the depiction of some of the vertices in more than one place, on the edge of the diagram. Unfortunately, the problem of determining the genus and finding such an embedding is [[http://en.wikipedia.org/wiki/NP-complete|NP-complete]]. However, finding bounds on the genus is a much easier problem. For very small scales, the genus routine of [[http://www.sagemath.org/|SAGE]] can manage to finish computing after an acceptable amount of time.

If the genus is g, the graph can also be drawn on any surface of genus greater than g; one way to accomplish that, but not the only way, is simply not to use more than g of the holes--that is, not to draw an edge passing through it. In fact, beyond a certain point you would not be able to use all of the holes and would have to leave some without edges drawn through them. The maximum number of holes, all of which can actually be used by having an edge pass through them, is called the maximal genus an is another invariant of the graph.

If the genus is g, the graph can also be drawn on any surface of genus greater than g; one way to accomplish that, but not the only way, is simply not to use more than g of the holes--that is, not to draw an edge passing through it. In fact, beyond a certain point you would not be able to use all of the holes and would have to leave some without edges drawn through them. The maximum number of holes, all of which can actually be used by having an edge pass through them, is called the maximal genus and is another invariant of the graph.

A 5-limit JI scale, meaning a scale composed of 5-limit JI intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the [[http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif|hexagonal lattice]] of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)

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Drawing (embedding) the graph on a surface with no crossings would give a nice visual picture of the harmonic relations in a scale. Moreover, such an embedding can be converted into a planar diagram by allowing the depiction of some of the vertices in more than one place, on the edge of the diagram. Unfortunately, the problem of determining the genus and finding such an embedding is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/NP-complete" rel="nofollow">NP-complete</a>. However, finding bounds on the genus is a much easier problem. For very small scales, the genus routine of <a class="wiki_link_ext" href="http://www.sagemath.org/" rel="nofollow">SAGE</a> can manage to finish computing after an acceptable amount of time.

If the genus is g, the graph can also be drawn on any surface of genus greater than g; one way to accomplish that, but not the only way, is simply not to use more than g of the holes--that is, not to draw an edge passing through it. In fact, beyond a certain point you would not be able to use all of the holes and would have to leave some without edges drawn through them. The maximum number of holes, all of which can actually be used by having an edge pass through them, is called the maximal genus an is another invariant of the graph.

If the genus is g, the graph can also be drawn on any surface of genus greater than g; one way to accomplish that, but not the only way, is simply not to use more than g of the holes--that is, not to draw an edge passing through it. In fact, beyond a certain point you would not be able to use all of the holes and would have to leave some without edges drawn through them. The maximum number of holes, all of which can actually be used by having an edge pass through them, is called the maximal genus and is another invariant of the graph.

A 5-limit JI scale, meaning a scale composed of 5-limit JI intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the <a class="wiki_link_ext" href="http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif" rel="nofollow">hexagonal lattice</a> of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)

@@ 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:keenanpepper|keenanpepper]] and made on <tt>2012-09-21 12:44:04 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-21 15:17:03 UTC</tt>.<br>
-: The original revision id was <tt>366711130</tt>.<br>
+: The original revision id was <tt>366763582</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 38: / Line 38: @@
 Drawing (embedding) the graph on a surface with no crossings would give a nice visual picture of the harmonic relations in a scale. Moreover, such an embedding can be converted into a planar diagram by allowing the depiction of some of the vertices in more than one place, on the edge of the diagram. Unfortunately, the problem of determining the genus and finding such an embedding is [[http://en.wikipedia.org/wiki/NP-complete|NP-complete]]. However, finding bounds on the genus is a much easier problem. For very small scales, the genus routine of [[http://www.sagemath.org/|SAGE]] can manage to finish computing after an acceptable amount of time.
-If the genus is g, the graph can also be drawn on any surface of genus greater than g; one way to accomplish that, but not the only way, is simply not to use more than g of the holes--that is, not to draw an edge passing through it. In fact, beyond a certain point you would not be able to use all of the holes and would have to leave some without edges drawn through them. The maximum number of holes, all of which can actually be used by having an edge pass through them, is called the maximal genus an is another invariant of the graph.
+If the genus is g, the graph can also be drawn on any surface of genus greater than g; one way to accomplish that, but not the only way, is simply not to use more than g of the holes--that is, not to draw an edge passing through it. In fact, beyond a certain point you would not be able to use all of the holes and would have to leave some without edges drawn through them. The maximum number of holes, all of which can actually be used by having an edge pass through them, is called the maximal genus and is another invariant of the graph.
 A 5-limit JI scale, meaning a scale composed of 5-limit JI intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the [[http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif|hexagonal lattice]] of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)
@@ Line 188: / Line 188: @@
 Drawing (embedding) the graph on a surface with no crossings would give a nice visual picture of the harmonic relations in a scale. Moreover, such an embedding can be converted into a planar diagram by allowing the depiction of some of the vertices in more than one place, on the edge of the diagram. Unfortunately, the problem of determining the genus and finding such an embedding is &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/NP-complete" rel="nofollow"&gt;NP-complete&lt;/a&gt;. However, finding bounds on the genus is a much easier problem. For very small scales, the genus routine of &lt;a class="wiki_link_ext" href="http://www.sagemath.org/" rel="nofollow"&gt;SAGE&lt;/a&gt; can manage to finish computing after an acceptable amount of time.&lt;br /&gt;
 &lt;br /&gt;
-If the genus is g, the graph can also be drawn on any surface of genus greater than g; one way to accomplish that, but not the only way, is simply not to use more than g of the holes--that is, not to draw an edge passing through it. In fact, beyond a certain point you would not be able to use all of the holes and would have to leave some without edges drawn through them. The maximum number of holes, all of which can actually be used by having an edge pass through them, is called the maximal genus an is another invariant of the graph.&lt;br /&gt;
+If the genus is g, the graph can also be drawn on any surface of genus greater than g; one way to accomplish that, but not the only way, is simply not to use more than g of the holes--that is, not to draw an edge passing through it. In fact, beyond a certain point you would not be able to use all of the holes and would have to leave some without edges drawn through them. The maximum number of holes, all of which can actually be used by having an edge pass through them, is called the maximal genus and is another invariant of the graph.&lt;br /&gt;
 &lt;br /&gt;
 A 5-limit JI scale, meaning a scale composed of 5-limit JI intervals such that {6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the consonance set, is always planar. This because the &lt;a class="wiki_link_ext" href="http://xenharmonic.wikispaces.com/file/detail/hexagonal-lattice.gif" rel="nofollow"&gt;hexagonal lattice&lt;/a&gt; of 5-limit pitch classes is planar, and the graph of any scale will simply be a finite subgraph of that lattice, considered as a graph. On the other hand, a non-contorted 5-limit scale in any equal temperament will be of genus 1, since the two commas serve as boundaries for a parallelogram which defines a torus (we can visualize this as rolling it up and glueing ends together.)&lt;br /&gt;