Convex scale
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 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. 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 for many scales that are the convex closures of interesting sets of pitches.
Formal definition
The following definitions make sense in the context of any Z-module, which is the same concept as an abelian group.
Convex combination
A convex combination of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that
[math]$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$[/math]
Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the injective hull of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by
[math]$c = c_1 + c_2 + \dots + c_k$[/math]
we obtain
[math]$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$[/math]
where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.
Convex set
A convex set (sometimes called a Z-polytope) in a lattice is a set that includes all convex combinations of its elements. The convex closure of a set of lattice elements is the smallest convex set containing the set.
Examples
- Every MOS is convex.
- In fact, every distributionally even scale is convex.
- Every Fokker block is convex.
- Every untempered tonality diamond is convex.
- Gallery of Z-polygon transversals
