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 of the template MOS is the number of rows of the parallelogram parallel to the unique step size parallel to a side of the parallelogram.

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

Step 1: Get a surjective homomorphism [math]\displaystyle{ \mathbb{Z}^2 \to \mathbb{Z}/mn\mathbb{Z} }[/math]

The π-image of any k-step interval (abelianized slice) ab(w[i : i + k]) is identical to that of ab(w[i : i + k + mn]). Hence there is a well-defined map from the pitch classes of intervals of w to ℤ/mnℤ. Traversing w step by step gives a traversal of [0 : m] × [0 : n]. We thus wish to constrain ways of labeling [0 : m] × [0 : n], with ℤ/mnℤ elements such that

• v = (1, 0) is consistently the π-image of a k_v-step interval of w, 0 < k_v < mn
• w = (0, 1) is consistently the π-image of a k_w-step interval, 0 < k_w < mn.