# 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