Parallelogram substring scale

A quasi-parallelogram scale is a scale whose pitch class lattice forms either a parallelogram or an incomplete traversal of one. Such scales are significant in rank-3 generator-offset theory.

Mathematical definition

An e-equivalent scale is a quasi-parallelogram if there exist integers m > 0, n > 0, 0 ≤ a < n, 0 ≤ b < n, a vector a, and two linearly independent vectors v and w such that the set of notes in the scale as a subset of the lattice of e-equivalent pitches is

[math]\displaystyle{ \{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1} \cup \{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1} \cup \{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}. % prefix of last row }[/math]

Here the scale is thought as traversing a series of rows one step of the row at a time, and

• [math]\displaystyle{ \{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1} }[/math] is a (nonempty) suffix of the first row
• [math]\displaystyle{ \{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1} }[/math] is a (possibly empty) parallelogram where rows are traversed fully
• [math]\displaystyle{ \{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b} }[/math] is a (nonempty) prefix of the last row
• v and w are the generator and offset

Ternary scales with this property

• All non-Fraenkel balanced MV3 scales
• All ax(by(a - b)z) MOS substitution scales
• All MOS substitution scales where:
  • The template MOS is primitive
  • There exists a pair (g, h) where:
    1. g is a generator of the template MOS
    2. h is a generator of the filling MOS
    3. |g|_X = |h| where X is the slot letter of the template MOS

Open problems

1. Classify all MOS-substitution quasi-parallelogram scales.
2. Classify all ternary quasi-parallelogram scales. Conjecture: All ternary quasi-parallelogram scales are MOS substitution scales.