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.

Hence MOS substitution scales with a primitive template MOS satisfy a property that we call ''almost parallelogram''{{~~idiosyncratic~~}}. 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

Hence MOS substitution scales with a primitive template MOS satisfy a property that we call ''almost parallelogram''{{User:Inthar/Template:adhoc}}. 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>

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# Is there a simple answer for when a MOS substitution scale becomes a MOS after deleting the "added" steps?

# For an arbitrary ternary scale that results from MOS substitution, when are the GSes obtained via the procedure the shortest possible GSes?

# Call an almost parallelogram scale with ''a'' = 1 and ''b'' = ''n'' − 2 ''transposable''{{~~idiosyncratic~~}}. Classify transposable MOS substitution scales.

# Call an almost parallelogram scale with ''a'' = 1 and ''b'' = ''n'' − 2 ''transposable''{{User:Inthar/Template:adhoc}}. Classify transposable MOS substitution scales.

[[Category:Math]]

@@ Line 152: / Line 152: @@
 * 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.
-Hence MOS substitution scales with a primitive template MOS satisfy a property that we call ''almost parallelogram''{{idiosyncratic}}. 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
+Hence MOS substitution scales with a primitive template MOS satisfy a property that we call ''almost parallelogram''{{User:Inthar/Template:adhoc}}. 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>
@@ Line 163: / Line 163: @@
 # Is there a simple answer for when a MOS substitution scale becomes a MOS after deleting the "added" steps?
 # For an arbitrary ternary scale that results from MOS substitution, when are the GSes obtained via the procedure the shortest possible GSes?
-# Call an almost parallelogram scale with ''a'' = 1 and ''b'' = ''n'' &minus; 2 ''transposable''{{idiosyncratic}}. Classify transposable MOS substitution scales.
+# Call an almost parallelogram scale with ''a'' = 1 and ''b'' = ''n'' &minus; 2 ''transposable''{{User:Inthar/Template:adhoc}}. Classify transposable MOS substitution scales.
 [[Category:Math]]