MOS substitution: Difference between revisions

In the original aberrismic-informed context, say that <math>d = (a, c) > 1.</math> Consider the MOS word <math>(a + c)\mathbf{X}b\mathbf{m}</math>, which we call the ''template MOS''. Since the "most even" arrangement (in the sense of [[distributional evenness]]) of <math>a</math>-many <math>\mathbf{L}</math> steps and <math>c</math>-many <math>\mathbf{s}</math> steps is the MOS <math>a\mathbf{L}b\mathbf{s}</math> (which will in general be a non-[[primitive]] MOS), this method prescribes following the latter MOS, called the ''filling MOS'', to fill in the <math>\mathbf{X}</math> steps. Fixing a choice of which <math>\mathbf{X}</math> in the MOS <math>(a + c)\mathbf{X}b\mathbf{m}</math> you start from, we can choose one of <math>(a+c)/d</math> modes of <math>a \mathbf{L} c \mathbf{s}.</math> If <math>a = c</math>, we obtain a balanced (thus MV3) ternary scale; when in addition <math>b</math> is odd, the scale is also SV3 and chiral, and we recover the two chiralities from the two modes of <math>a\mathbf{L}c\mathbf{s}</math>. Of course, one may do this using template MOS <math>a\mathbf{L}(b + c)\mathbf{X}</math> and the <math>(b, c)</math>-multiperiod filling MOS <math>b\mathbf{m} c\mathbf{s}</math> instead. We tentatively denote the resulting scale <math>\mathsf{insert\_by\_MOS\_subst}(a, b, c; x; k),</math> where <math>x \in \{\mathbf{L}, \mathbf{m}\}</math> is the step size identified with <math>\mathbf{s}</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>).

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

To derive <math>5\mathbf{L}2\mathbf{m}4\mathbf{s}</math> as <math>\mathsf{insert\_by\_MOS\_subst}(5, 2, 4; \mathbf{m}; 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 short 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:

|+ <math>5\mathbf{L}2\mathbf{m}4\mathbf{s}</math>  as <math>\mathsf{insert\_by\_MOS\_subst}(5, 2, 4; \mathbf{m}; k)</math>

|+ <math>6\mathbf{L}7\mathbf{m}9\mathbf{s}</math> as <math>\mathsf{insert\_by\_MOS\_subst}(6, 7, 9; \mathbf{L}; k)</math>