Graph-theoretic properties of scales - Revision history

Sintel: -legacy

2025-03-29T18:00:45Z

-legacy

← Older revision		Revision as of 18:00, 29 March 2025
Line 1:		Line 1:
	~~{{Legacy}}~~
	[[Periodic scale]]s can be modeled as [[wikipedia:Graph (discrete mathematics)\|graph]]s in the sense of {{w\|graph theory}}. By applying concepts of graph theory to scales, it is possible to analyze their structure, from which some musical properties can be derived.		[[Periodic scale]]s can be modeled as [[wikipedia:Graph (discrete mathematics)\|graph]]s in the sense of {{w\|graph theory}}. By applying concepts of graph theory to scales, it is possible to analyze their structure, from which some musical properties can be derived.

Fredg999: Add lead section, cleanup (heading levels, sentence case in headings, simplify internal links, use Template:W, etc.), categories

2025-02-21T03:34:12Z

Add lead section, cleanup (heading levels, sentence case in headings, simplify internal links, use Template:W, etc.), categories

Show changes

Lériendil: Undo revision 181054 by Lériendil (talk)

2025-02-17T16:52:43Z

Undo revision 181054 by Lériendil (talk)

← Older revision		Revision as of 16:52, 17 February 2025
Line 1:		Line 1:
	{{Legacy~~\|Graph-theoretic_properties_of_scales~~}}		{{Legacy}}
	__FORCETOC__		__FORCETOC__
	=Graph of a scale=		=Graph of a scale=

Lériendil at 16:05, 17 February 2025

2025-02-17T16:05:47Z

← Older revision		Revision as of 16:05, 17 February 2025
Line 1:		Line 1:
	{{Legacy}}		{{Legacy\|Graph-theoretic_properties_of_scales}}
	__FORCETOC__		__FORCETOC__
	=Graph of a scale=		=Graph of a scale=

Lériendil: deploying new cat

2025-02-17T16:05:26Z

deploying new cat

← Older revision		Revision as of 16:05, 17 February 2025
Line 1:		Line 1:
			{{Legacy}}
	__FORCETOC__		__FORCETOC__
	=Graph of a scale=		=Graph of a scale=

Lériendil: Changed protection level for "Graph-theoretic properties of scales" ([Edit=Allow only administrators] (expires 16:59, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 16:59, 17 February 2025 (UTC)))

2025-02-17T15:59:53Z

Changed protection level for "Graph-theoretic properties of scales" ([Edit=Allow only administrators] (expires 16:59, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 16:59, 17 February 2025 (UTC)))

← Older revision	Revision as of 15:59, 17 February 2025
(No difference)

Mike Battaglia: Protected "Graph-theoretic properties of scales" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:01:15Z

Protected "Graph-theoretic properties of scales" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:01, 15 November 2021
(No difference)

Cmloegcmluin: remove temporary link to defactoring

2021-11-12T23:01:25Z

remove temporary link to defactoring

← Older revision		Revision as of 23:01, 12 November 2021
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.		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:		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:

Cmloegcmluin: /* The Genus */ add link to defactored

2021-09-29T19:41:26Z

The Genus: add link to defactored

← Older revision		Revision as of 19:41, 29 September 2021
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.		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 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]] (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 [[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:		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:

Spt3125: external image replaced with local copy

2018-09-22T02:26:21Z

external image replaced with local copy

← Older revision		Revision as of 02:26, 22 September 2018
Line 119:		Line 119:
	The first note of any of the 16 tetrads of Star can be either 0 or 1, the second 2 or 3, the third 4 or 5, and the fourth 6 or 7. Any of the resulting 16 possible combinations will not have the forbidden intervals (0, 1), (2, 3), (4, 5), or (6, 7) which are too small to give 11-limit consonances.		The first note of any of the 16 tetrads of Star can be either 0 or 1, the second 2 or 3, the third 4 or 5, and the fourth 6 or 7. Any of the resulting 16 possible combinations will not have the forbidden intervals (0, 1), (2, 3), (4, 5), or (6, 7) which are too small to give 11-limit consonances.

	[[File:star.png\|alt=star.png\|star.png]]		[[File:star.png\|alt=star.png\|star.png]] [[file:16-cell.gif]]

	~~<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right~~:~~0.5em; display:inline-block'>~~
	~~External image: http://upload.wikimedia.org/wikipedia/commons/a/a0/~~16-cell.gif~~<br>~~
	~~: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>~~
	~~: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>~~
	~~</div>~~
	~~[[Category:IMPORTDEBUG - Change External Images~~]]

	[http://micro.soonlabel.com/star/20120830-77et-star.mp3 77et Star] by [[Chris_Vaisvil\|Chris Vaisvil]]		[http://micro.soonlabel.com/star/20120830-77et-star.mp3 77et Star] by [[Chris_Vaisvil\|Chris Vaisvil]]