Ternary parallelogram scales are MOS substitution: Difference between revisions
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== Proof == | == Proof == | ||
=== Step 1: Get a surjective homomorphism <math>\mathbb{Z}^2 \to \mathbb{Z}/mn\mathbb{Z}</math> === | === Step 1: Get a surjective homomorphism <math>\mathbb{Z}^2 \to \mathbb{Z}/mn\mathbb{Z}</math> === | ||
The π-image of any ''k''-step interval (abelianized slice) {{nowrap|ab(''w''[''i'' : ''i'' + ''k''])}} is identical to that of {{nowrap|ab(''w''[''i'' : ''i'' + ''k'' + ''mn'']).}} Hence there is a well-defined map from pitch-class group elements to {{nowrap|ℤ/''mn''ℤ.}} We thus wish to constrain ways of labeling {{nowrap|[0 : ''m''] × [0 : ''n''],}} an {{nowrap|''m'' × ''n''}} grid, with {{nowrap|ℤ/''mn''ℤ}} elements such that | |||
* {{nowrap|'''v''' {{=}} (1, 0)}} is consistently the π-image of a ''k''<sub>'''v'''</sub>-step interval of ''w'' | |||
* {{nowrap|'''w''' {{=}} (0, 1)}} is consistently the π-image of a ''k''<sub>'''w'''</sub>-step interval. | |||
=== Step 2: By ternarity, exactly one of the step vectors is parallel to a coordinate axis === | === Step 2: By ternarity, exactly one of the step vectors is parallel to a coordinate axis === | ||