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 the pitch classes of intervals of ''w'' to {{nowrap|ℤ/''mn''ℤ.}} Traversing ''w'' step by step gives a traversal of {{nowrap|[0 : ''m''] × [0 : ''n'']}} where we label each grid point with the index of the current note in ''w''. We thus wish to constrain ways of labeling {{nowrap|[0 : ''m''] × [0 : ''n'']}} with {{nowrap|ℤ/''mn''ℤ}} elements such that | 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 the pitch classes of intervals of ''w'' to {{nowrap|ℤ/''mn''ℤ.}} Traversing ''w'' step by step gives a traversal of {{nowrap|[0 : ''m''] × [0 : ''n'']}} where we label each grid point with the index of the current note in ''w''. We also recall that the pitch-class vector '''v''' has a representative that is a ''k''<sub>'''v'''</sub>-step interval in ''w'', {{nowrap|0 < ''k''<sub>'''v'''</sub> < ''mn'',}} and similarly for '''w'''. | ||
We thus wish to constrain ways of labeling {{nowrap|[0 : ''m''] × [0 : ''n'']}} 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|0 < ''k''<sub>'''v'''</sub> < ''mn''}} | * {{nowrap|'''v''' {{=}} (1, 0)}} is consistently the π-image of a ''k''<sub>'''v'''</sub>-step interval of ''w'', {{nowrap|0 < ''k''<sub>'''v'''</sub> < ''mn''}} | ||
* {{nowrap|'''w''' {{=}} (0, 1)}} is consistently the π-image of a ''k''<sub>'''w'''</sub>-step interval, {{nowrap|0 < ''k''<sub>'''w'''</sub> < ''mn''}} | * {{nowrap|'''w''' {{=}} (0, 1)}} is consistently the π-image of a ''k''<sub>'''w'''</sub>-step interval, {{nowrap|0 < ''k''<sub>'''w'''</sub> < ''mn''}} | ||