Parallelogram substring scale: Difference between revisions
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A '''quasi-parallelogram scale''' is a scale whose pitch class lattice forms either a parallelogram or an incomplete traversal of one. | A '''quasi-parallelogram scale''' is a scale whose pitch class lattice forms either a parallelogram or an incomplete traversal of one. They are significant in rank-3 generator-offset theory. | ||
== Mathematical definition == | == 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 | 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 | ||