MOS substitution: Difference between revisions

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Here <math>\mathbf{z}</math> is the new step size inserted, <math>\mathbf{y}</math> is the step size in the starting MOS identified with <math>\mathbf{z}</math> by the template MOS, and <math>k</math> is the brightness of the mode of the filling MOS used (<math>k = 0</math> corresponds to the darkest mode; the conventional understanding of "brightness" makes sense as <math>\mathbf{L}</math> (resp. <math>\mathbf{m}</math>) > <math>\mathbf{s}</math>).

== Denoting modes of a MOS substitution scale ==

Just as rotating the filling MOS while fixing the mode of the template MOS serves to distinguish different MOS substitution scales, we take the mode of the template MOS (treating the slot letter as the smaller step) and write it in UDP to indicate the mode of a given MOS substitution scale.

== Examples ==

In the following tables, the interval class of the generators stacked in the generator sequence is such that the perfect generator has fewer <math>\mathbf{X}</math> steps than the imperfect counterpart.

=== 5L2m4s ===

To derive groundfault's [[diamech]] scale which has step pattern <math>5\mathbf{L}2\mathbf{m}4\mathbf{s}</math> as <math>\mathsf{MOS\_subst}(5, 2, 4; \mathbf{m}, \mathbf{s}; k)</math>, we exploit <math>(b, c) = 2</math> and substitute <math>2\mathbf{m}4\mathbf{s}</math> into the template MOS <math>5\mathbf{L}6\mathbf{X}</math> (<math>\mathbf{LXLXLXLXLXX}</math>). Since <math>2\mathbf{m}4\mathbf{s}</math> has three distinct modes (<math>\mathbf{ssmssm}, \mathbf{smssms}, \mathbf{mssmss}</math>) and <math>5\mathbf{L}6\mathbf{X}</math> is primitive, we obtain three distinct scales, all of which admit length-3 generator sequences of 2-steps, representing all 3 possible rotations of <math>(\mathbf{L}+\mathbf{m}, \mathbf{L}+\mathbf{s}, \mathbf{L}+\mathbf{s})</math> as displayed in the following table:

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=== 5L2m6s ===

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Here the notation ''G''^''k'' denotes repeating the generator ''G'' ''k'' times in the generator sequence.

These are four of the 8 [[billiard scale]]s that have pattern 5'''L'''2'''m'''6'''s'''. The other four billiard words have length-3 subwords of non-'''X''' letters, unlike the MOS substitution scales.

=== 6L7m9s ===

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== Mathematical facts ==

If a ternary scale with [[step signature]] ''a'''''L'''''b'''''m'''''c'''''s''' satisfies:

# the result of identifying '''L''' steps with '''m''' steps is a MOS;

# the result of deleting all '''s''' steps is a MOS,

=== If the template MOS is primitive, MOS substitution yields binary well-formed generator sequences ===

Consider the mode of the template MOS <math>T = T(\mathbf{m},\mathbf{X}) = (a+c)\mathbf{X}b\mathbf{m}(0).</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>a+c,</math> <math>r</math>-steps in the filling MOS <math>F = a\mathbf{L}c\mathbf{s}(k)</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>(a+c - 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.

With the additional assumption that the number of X's in a perfect generator ''p''_''T'' of the template MOS be a generator class of the filling MOS, the generator sequence yields ''q'' parallel chains ''C''₁,  

..., ''C''_''q'' of the aggregate generator. The offset between ''C''_''i'' and ''C''_''i''+1 is equal to subst(''p''_''T'', '''X''', ''p''_''F''), where ''p''_''T'' and ''p''_''F'' are perfect generators (of appropriate lengths) of the template and filling MOSes, respectively. The aggregate generator is  subst((''p''_''T'')^''r'', '''X''', ''F''^''r''), where ''F'' is the filling MOS.

=== MOS substitution scales have block balance at most 2 ===

== MOS substitution scales and RTT ==

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