MOS substitution: Difference between revisions
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=== If the template MOS is primitive, MOS substitution yields almost parallelograms in the lattice === | === If the template MOS is primitive, MOS substitution yields almost parallelograms in the lattice === | ||
The generator sequence thus yields ''q'' parallel chains ''C''<sub>1</sub>, | The generator sequence thus yields ''q'' parallel chains ''C''<sub>1</sub>, | ||
..., ''C''<sub>''q''</sub> of the aggregate generator | ..., ''C''<sub>''q''</sub> of the aggregate generator. The offset between ''C''<sub>''i''</sub> and ''C''<sub>''i''+1</sub> is equal to subst(''p''<sub>''t''</sub>, '''X''', ''p''<sub>''f''</sub>), where ''p''<sub>''t''</sub> and ''p''<sub>''f''</sub> are perfect generators (of the appropriate length) of the template and filling MOSes, respectively. The aggregate generator is subst((''p''<sub>''t''</sub>)<sup>''r''</sup>, '''X''', ''f''<sup>''r''</sup>), where ''f'' is the filling MOS and ''r'' is the number of steps in one generator of ''f''. | ||
Hence in the GS, | |||
* the perfect generator of the filling MOS corresponds to advancing from ''C''<sub>''i''</sub> to ''C''<sub>''i''+1</sub>; | * the perfect generator of the filling MOS corresponds to advancing from ''C''<sub>''i''</sub> to ''C''<sub>''i''+1</sub>; | ||
* the imperfect generator of the filling MOS corresponds to looping back to ''C''<sub>1</sub> but on the next note of ''C''<sub>1</sub>, so it and the ''q'' − 1 notes thereafter are advanced by 1 note from any predecessor notes in the chains. | * the imperfect generator of the filling MOS corresponds to looping back to ''C''<sub>1</sub> but on the next note of ''C''<sub>1</sub>, so it and the ''q'' − 1 notes thereafter are advanced by 1 note from any predecessor notes in the chains. | ||
Hence MOS substitution scales with a primitive template MOS satisfy a property that we call ''almost parallelogram''. An '''e'''-equivalent scale is ''almost a parallelogram'' if there exist non-negative integers ''m'', ''n'' | Hence MOS substitution scales with a primitive template MOS satisfy a property that we call ''almost parallelogram''. An '''e'''-equivalent scale is ''almost a parallelogram'' if there exist non-negative integers ''m'', ''n'', 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 | ||
<math>\{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1} \cup \{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1} \cup \{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}.</math> | <math>\{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1} \cup \{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1} \cup \{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}.</math> | ||
In the above case, ''n'' = ''q'', '''v''' = subst(''P''<sub>''T''</sub>, '''X''', ''P''<sub>''F''</sub>), and '''w''' = subst((''p''<sub>''t''</sub>)<sup>''r''</sup>, '''X''', ''f''<sup>''r''</sup>). | |||
The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is almost a parallelogram. | The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is almost a parallelogram. | ||