Ternary parallelogram scales are MOS substitution

This article proves the following theorem:

Ternary parallelogram scale words are MOS substitution scale words, where the period count (gcd of step signature) of the template MOS is the number of rows of the parallelogram parallel to the unique "axial" step.

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 (equiv. a free abelian group) 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.

Proof