Convex scale: Difference between revisions
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{{Todo}}[[File:Lattice Marvel Convex12.png|400px|thumb|A convex set of 12 tones from the marvel lattice.]] | {{Todo|inline=1|expand|comment=explain musical application --[[User:Hkm|hkm]] ([[User talk:Hkm|talk]]) 20:39, 25 June 2025 (UTC)}}[[File:Lattice Marvel Convex12.png|400px|thumb|A convex set of 12 tones from the marvel lattice.]] | ||
In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' (also called a Z-polytope) 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''' (also called a Z-polytope) 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. | ||
Latest revision as of 20:39, 25 June 2025
In a regular temperament, a convex scale is a set of pitches that form a convex set (also called a Z-polytope) in the interval lattice of the temperament. The "regular temperament" is often 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 convex region of continuous space. Alternatively, a convex set in a lattice is a set where any weighted average of elements (where no element has negative weight) is within the set if it is on the lattice.
The convex hull or convex closure of a scale is the smallest convex scale that contains it. (Every scale has a unique convex hull.) See Gallery of Z-polygon transversals for many scales that are the convex closures of interesting sets of pitches.
Examples
- Every MOS is convex.
- In fact, every distributionally even scale is convex.
- Every Fokker block is convex.
- Every untempered tonality diamond is convex.