Graph-theoretic properties of scales: Difference between revisions

Line 33:

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 [[:File: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]] ~~(or in other words, [[defactored]])~~ 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.)

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 [[:File: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.)

A [[Dyadic_chord|dyadic chord]] pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below:

@@ Line 33: / Line 33: @@
 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 [[:File: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]] (or in other words, [[defactored]]) 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.)
+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 [[:File: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.)
 A [[Dyadic_chord|dyadic chord]] pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below: