Ternary parallelogram scales are MOS substitution: Difference between revisions
Created page with "This article proves the following theorem: ''Primitive ternary parallelogram scale words are MOS substitution scale words.'' == Definitions == === Pitch-class group === The ''pitch-class group'' of a scale word ''w'' in letters {{nowrap|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}} with step signature {{nowrap|'''s''' ∈ ℤ<sup>''r''</sup>{{angbr|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}}}} is the abelian group {{nowrap|C(''w'') :{{=}} ℤ<sup>'..." |
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The ''pitch-class group'' of a scale word ''w'' in letters {{nowrap|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}} with [[step signature]] {{nowrap|'''s''' ∈ ℤ<sup>''r''</sup>{{angbr|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}}}} is the abelian group {{nowrap|C(''w'') :{{=}} ℤ<sup>''r''</sup>{{angbr|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}}/{{angbr|'''s'''}}.}} The pitch-class group is associated with a canonical map π that sends every step vector to its pitch class. | The ''pitch-class group'' of a scale word ''w'' in letters {{nowrap|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}} with [[step signature]] {{nowrap|'''s''' ∈ ℤ<sup>''r''</sup>{{angbr|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}}}} is the abelian group {{nowrap|C(''w'') :{{=}} ℤ<sup>''r''</sup>{{angbr|'''x'''<sub>1</sub>, ..., '''x'''<sub>''r''</sub>}}/{{angbr|'''s'''}}.}} The pitch-class group is associated with a canonical map π that sends every step vector to its pitch class. | ||
=== Parallelogram scale === | === Parallelogram scale === | ||
A scale word ''w'' in is a ''parallelogram scale word'' if there exists integers {{nowrap|''m'', ''n'' > 1}} and nonzero elements '''v''' and '''w''' in C(''w'') such that the π-image of | A scale word ''w'' in is a ''parallelogram scale word'' if C(''w'') is torsion-free and there exists integers {{nowrap|''m'', ''n'' > 1}} and nonzero elements '''v''' and '''w''' in C(''w'') such that the π-image of | ||
<math>\mathcal{I}_w := | <math>\mathcal{I}_w := | ||
Revision as of 17:57, 14 March 2026
This article proves the following theorem:
Primitive 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, ..., xr with step signature s ∈ ℤr⟨x1, ..., xr⟩ is the abelian group C(w) := ℤr⟨x1, ..., xr⟩/⟨s⟩. 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 in is a parallelogram scale word if C(w) is torsion-free and there exists integers m, n > 1 and nonzero 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.