Ternary parallelogram scales are MOS substitution

This article proves the following theorem:

Ternary parallelogram scale words are MOS substitution scale words.

Definitions

Pitch-class group

The pitch-class group of a scale word w in letters x₁, ..., x_r with step signature e ∈ ℤ^r⟨x₁, ..., x_r⟩ is the abelian group C(w) := ℤ^r⟨x₁, ..., x_r⟩/⟨e⟩. The pitch-class group is associated with a canonical map π that sends every step vector to its pitch class.

Parallelogram scale

A scale word w is a parallelogram scale word if C(w) is torsion-free and there exists integers m, n > 1 and linearly independent elements v and w in C(w) such that the π-image of

[math]\displaystyle{ \mathcal{I}_w := \{\mathrm{ab}(\epsilon), \mathrm{ab}(w[0:1]), ..., \mathrm{ab}(w[0:|w|-1])\} }[/math]

is of the form

[math]\displaystyle{ \{i\mathbf{v} + j\mathbf{w} : i \in [0:m], j \in [0:n]\}. }[/math]

MOS substitution scale

See MOS substitution.