Ternary parallelogram scales are MOS substitution: Difference between revisions
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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 C(''w'') is torsion-free and there exists integers {{nowrap|''m'', ''n'' > 1}} and | 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 linearly independent elements '''v''' and '''w''' in C(''w'') such that the π-image of | ||
<math>\mathcal{I}_w := | <math>\mathcal{I}_w := | ||