Convex scale: Difference between revisions

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In a [[~~Regular_Temperaments|~~regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[~~Just_intonation~~|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [http://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.

The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[~~Gallery_of_Z-polygon_transversals|~~Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.

The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.

==Formal definition==

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==Examples==

~~<ul><li>~~Every [[MOSScales|MOS]] is convex.~~</li><li>~~In fact, every [[~~distributionally_even|~~distributionally even]] scale is convex.~~</li><li>~~Every [[~~Fokker_blocks|~~Fokker block]] is convex.~~</li><li>~~Every untempered [[~~Tonality_diamond|~~tonality diamond]] is convex.~~</li><li>~~[[~~Gallery_of_Z-polygon_transversals|~~Gallery of Z-polygon transversals]]~~</li></ul> [[Category:math]]~~

[[Category:~~scale~~]]

* Every [[MOSScales|MOS]] is convex.

[[Category:theory]]

*In fact, every [[distributionally even]] scale is convex.

* Every [[Fokker block]] is convex.

* Every untempered [[tonality diamond]] is convex.

* [[Gallery of Z-polygon transversals]]

[[Category:Math]]

[[Category:Scale theory]]

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-In a [[Regular_Temperaments|regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just_intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.
+In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.
 A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [http://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.
-The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery_of_Z-polygon_transversals|Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.
+The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.
 ==Formal definition==
@@ Line 27: / Line 27: @@
 ==Examples==
-<ul><li>Every [[MOSScales|MOS]] is convex.</li><li>In fact, every [[distributionally_even|distributionally even]] scale is convex.</li><li>Every [[Fokker_blocks|Fokker block]] is convex.</li><li>Every untempered [[Tonality_diamond|tonality diamond]] is convex.</li><li>[[Gallery_of_Z-polygon_transversals|Gallery of Z-polygon transversals]]</li></ul>      [[Category:math]]
-[[Category:scale]]
+* Every [[MOSScales|MOS]] is convex.
-[[Category:theory]]
+*In fact, every [[distributionally even]] scale is convex.
+* Every [[Fokker block]] is convex.
+* Every untempered [[tonality diamond]] is convex.
+* [[Gallery of Z-polygon transversals]]
+[[Category:Math]]
+[[Category:Scale theory]]