MOS substitution: Difference between revisions

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* the perfect generator of the filling MOS corresponds to advancing from ''C''''i'' to ''C''''i''+1;

* the imperfect generator of the filling MOS corresponds to looping back to ''C''1 but on the next note of ''C''1, so it and the ''q'' − 1 notes thereafter are advanced by 1 note from any predecessor notes in the chains.

In other words, there exist non-negative integers ''m'', ''n'', ''a'' < ''n'', ''b'' < ''n'', and vectors '''a''', '''v''' and '''w''' in the lattice such that the set of notes in the scale 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-1]_1} \cup \{\mathbf{a} + i\mathbf{v} + m\mathbf{w}\}_{i=0}^{b}.</math>

Let us call rank-3 scales with such lattice shapes ''almost parallelogram-shaped''. MOS substitution scales are thus almost parallelogram-shaped, but 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 parallelogram-shaped.

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 * 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'' &minus; 1 notes thereafter are advanced by 1 note from any predecessor notes in the chains.
+In other words, there exist non-negative integers ''m'', ''n'', ''a'' < ''n'', ''b'' < ''n'', and vectors '''a''', '''v''' and '''w''' in the lattice such that the set of notes in the scale 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-1]_1} \cup \{\mathbf{a} + i\mathbf{v} + m\mathbf{w}\}_{i=0}^{b}.</math>
 Let us call rank-3 scales with such lattice shapes ''almost parallelogram-shaped''. MOS substitution scales are thus almost parallelogram-shaped, but 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 parallelogram-shaped.