Convex scale
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.