MOS substitution: Difference between revisions

Line 121:

== Facts ==

=== MOS substitution yields generator sequences ===

The following holds for <math>S = \mathsf{MOS\_subst}(a, b, c; \mathbf{L}, \mathbf{s}; k)</math> (and after switching <math>\mathbf{L}</math> with <math>\mathbf{m}</math> and <math>a</math> with <math>b,</math> for <math>\mathsf{MOS\_subst}(a, b, c; \mathbf{m}, \mathbf{s}; k)</math> as well):

Line 126:

Line 127:

Consider the mode of the template MOS <math>T = T(\mathbf{m},\mathbf{X}) = \mathsf{MOS}(a+c,b;0)(\mathbf{X},\mathbf{m}).</math> This is the mode of <math>T</math> that has the most <math>\mathbf{X}</math> steps near the end. If <math>T</math> is [[primitive]], let <math>r</math> be the count of <math>\mathbf{X}</math> steps in a chosen (reduced) generator of <math>T.</math> Since <math>r</math> must be coprime to <math>n,</math> <math>r</math>-steps in the filling MOS <math>F = \mathsf{MOS}(a,c;k)(\mathbf{L},\mathbf{s})</math> come in exactly 2 sizes, <math>i\mathbf{L}+j\mathbf{s}</math> and <math>(i-1)\mathbf{L}+(j+1)\mathbf{s}.</math> Since the detempering of the imperfect generator of <math>T</math> occurs only once in <math>S</math>, <math>S</math> admits a particularly elegant well-formed binary (using two distinct generators) [[generator sequence]] of length <math>q,</math> corresponding to the circle of <math>r</math>-steps in the filling MOS. Letting <math>\mathsf{GS}(g_1, ..., g_{q})</math> be this generator sequence, <math>g_j</math> is either <math>p\mathbf{m} + i\mathbf{L} + j\mathbf{s}</math> or <math>p\mathbf{m} + (i-1)\mathbf{L} + (j+1)\mathbf{s},</math> according as the <math>j</math>-th <math>r</math>-step in the sequence of stacked <math>r</math>-steps on the chosen mode of <math>F</math> is <math>i\mathbf{L} + j\mathbf{s}</math> or <math>(i-1)\mathbf{L} + (j+1)\mathbf{s}.</math> (We could have chosen to use the mode of <math>T</math> on the other extreme of its generator arc instead, which corresponds to taking the circle of <math>(n - r)</math>-steps in <math>F</math> and is thus also valid.) The generator of the template MOS serves as the "guide generator" for this generator sequence.

=== MOS substitution yields almost parallelograms in the lattice ===

The generator sequence thus yields ''q'' parallel chains ''C''1,

..., ''C''''q'' of the aggregate generator, and the offset between ''C''''i'' and ''C''''i''+1 is equal to the result of substituting the perfect generator of the filling MOS into the generator of the template MOS. Hence in the GS,

Line 132:

Line 133:

* 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'', 0 < ''a'' < ''n'', 0 < ''b'' < ''n'', and vectors '''a''', '''v''', and '''w''' in the lattice such that the set of notes in the scale is

In other words, there exist non-negative integers ''m'', ''n'', 0 < ''a'' < ''n'', 0 < ''b'' < ''n'', and vectors '''a''', '''v''', and '''w''' in the pitch-class 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~~.

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.

== Open questions ==

@@ Line 121: / Line 121: @@
 == Facts ==
+=== MOS substitution yields generator sequences ===
 The following holds for <math>S = \mathsf{MOS\_subst}(a, b, c; \mathbf{L}, \mathbf{s}; k)</math> (and after switching <math>\mathbf{L}</math> with <math>\mathbf{m}</math> and <math>a</math> with <math>b,</math> for <math>\mathsf{MOS\_subst}(a, b, c; \mathbf{m}, \mathbf{s}; k)</math> as well):
@@ Line 126: / Line 127: @@
 Consider the mode of the template MOS <math>T = T(\mathbf{m},\mathbf{X}) = \mathsf{MOS}(a+c,b;0)(\mathbf{X},\mathbf{m}).</math> This is the mode of <math>T</math> that has the most <math>\mathbf{X}</math> steps near the end. If <math>T</math> is [[primitive]], let <math>r</math> be the count of <math>\mathbf{X}</math> steps in a chosen (reduced) generator of <math>T.</math> Since <math>r</math> must be coprime to <math>n,</math> <math>r</math>-steps in the filling MOS <math>F = \mathsf{MOS}(a,c;k)(\mathbf{L},\mathbf{s})</math> come in exactly 2 sizes, <math>i\mathbf{L}+j\mathbf{s}</math> and <math>(i-1)\mathbf{L}+(j+1)\mathbf{s}.</math> Since the detempering of the imperfect generator of <math>T</math> occurs only once in <math>S</math>, <math>S</math> admits a particularly elegant well-formed binary (using two distinct generators) [[generator sequence]] of length <math>q,</math> corresponding to the circle of <math>r</math>-steps in the filling MOS. Letting <math>\mathsf{GS}(g_1, ..., g_{q})</math> be this generator sequence, <math>g_j</math> is either <math>p\mathbf{m} + i\mathbf{L} + j\mathbf{s}</math> or <math>p\mathbf{m} + (i-1)\mathbf{L} + (j+1)\mathbf{s},</math> according as the <math>j</math>-th <math>r</math>-step in the sequence of stacked <math>r</math>-steps on the chosen mode of <math>F</math> is <math>i\mathbf{L} + j\mathbf{s}</math> or <math>(i-1)\mathbf{L} + (j+1)\mathbf{s}.</math> (We could have chosen to use the mode of <math>T</math> on the other extreme of its generator arc instead, which corresponds to taking the circle of <math>(n - r)</math>-steps in <math>F</math> and is thus also valid.) The generator of the template MOS serves as the "guide generator" for this generator sequence.
+=== MOS substitution yields almost parallelograms in the lattice ===
 The generator sequence thus yields ''q'' parallel chains ''C''<sub>1</sub>,
 ..., ''C''<sub>''q''</sub> of the aggregate generator, and the offset between ''C''<sub>''i''</sub> and ''C''<sub>''i''+1</sub> is equal to the result of substituting the perfect generator of the filling MOS into the generator of the template MOS. Hence in the GS,
@@ Line 132: / Line 133: @@
 * 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'', 0 < ''a'' < ''n'', 0 < ''b'' < ''n'', and vectors '''a''', '''v''', and '''w''' in the lattice such that the set of notes in the scale is
+In other words, there exist non-negative integers ''m'', ''n'', 0 < ''a'' < ''n'', 0 < ''b'' < ''n'', and vectors '''a''', '''v''', and '''w''' in the pitch-class 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.
+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.
 == Open questions ==