MOS substitution: Difference between revisions

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* 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.

~~In other words,~~ there exist non-negative integers ''m'', ''n'' (in this case ''n'' = ''q''), 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 pitch-class lattice is

Hence MOS substitution scales satisfy a property that we call ''almost parallelogram''. A scale is ''almost a parallelogram'' if there exist non-negative integers ''m'', ''n'' (in this case ''n'' = ''q''), 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 pitch-class lattice 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>

~~Let us call rank-3 scales with such lattice shapes ''almost parallelograms''. MOS substitution scales are thus almost parallelograms, 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 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.

== Open questions ==

@@ Line 134: / Line 134: @@
 * 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'' (in this case ''n'' = ''q''), 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 pitch-class lattice is
+Hence MOS substitution scales satisfy a property that we call ''almost parallelogram''. A scale is ''almost a parallelogram'' if there exist non-negative integers ''m'', ''n'' (in this case ''n'' = ''q''), 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 pitch-class lattice 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>
-Let us call rank-3 scales with such lattice shapes ''almost parallelograms''. MOS substitution scales are thus almost parallelograms, 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 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.
 == Open questions ==