Parallelogram substring scale

A parallelogram substring scale is a scale whose pitch class lattice forms either a parallelogram or an incomplete traversal of one. Such scales are currently being investigated in ternary generator-offset theory.

Mathematical definition

An e-equivalent rank-3 scale is a parallelogram substring 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 is called the row generator. (-v would also satisfy the definition.)

This concept generalizes in the obvious way to arbitrary rank d (where each (d - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the parallelotope substring property.

A parallelogram substring scale with full first and last rows is a parallelogram scale.

Ternary scales with this property

Examples

• All non-Fraenkel balanced primitive MV3 scales
• All ax(by(a - b)z) MOS substitution scales if gcd(a, b) = 1
• 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

Non-examples

• All multiperiod MOS substitution scales (e.g. 4L(10m10s))

Mathematical facts

Ternary parallelogram scales are MOS substitution

Main article: Ternary parallelogram scales are MOS substitution

Open problems

1. Classify all MOS-substitution parallelogram substring scales.
2. Classify all ternary parallelogram substring scales.
   • Conjecture: All ternary parallelogram substring scales are MOS substitution scales. (Numerous counterexamples, e.g. LLmLmLmLmLLs)
3. Classify all ternary full parallelogram scales (PS with full first and last rows).