MOS substitution: Difference between revisions

Line 12:

Originally developed by Inthar for the purpose of adding [[aberrisma]] steps in an orderly manner to a MOS pattern <math>a\mathbf{L}b\mathbf{m}</math> (which we write in place of <math>a\mathbf{L}b\mathbf{s}</math> for convenience's sake, since <math>\mathbf{s}</math> denotes the new aberrisma steps added to the MOS) in the context of groundfault's [[aberrismic theory]], MOS substitution is intended to take advantage of extra potential symmetry when <math>a, c</math> or <math>b, c</math> is not a coprime pair and mildly generalize the congruence substitution procedure for building [[balanced]] words to obtain non-balanced but still more "even" scales with simple [[generator sequence]] expressions (in the sense of being binary, i.e. using only two distinct generators).

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. This article denotes the resulting scale <math>\mathsf{MOS\_subst}(a, b, c; \mathbf{x}, \mathbf{y}; k),</math> where <math>\mathbf{y}</math> is the new step size inserted, <math>\mathbf{x}</math> is the step size in the starting MOS identified with <math>\mathbf{y}</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>).

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. This article denotes the resulting scale <math>\mathsf{MOS\_subst}(a, b, c; \mathbf{y}, \mathbf{z}; k):</math>

<math>\displaystyle{ \mathsf{MOS\_subst}(a, b, c; \mathbf{y}, \mathbf{z}; k) := \mathsf{subst}( a\mathbf{w}(b + c)\mathbf{X}[0|{a+b+c-1}] , \mathbf{X}, b\mathbf{y}c\mathbf{z}[k|b+c-k-1] ) }</math>

where <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>).

==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.

@@ Line 12: / Line 12: @@
 Originally developed by Inthar for the purpose of adding [[aberrisma]] steps in an orderly manner to a MOS pattern <math>a\mathbf{L}b\mathbf{m}</math> (which we write in place of <math>a\mathbf{L}b\mathbf{s}</math> for convenience's sake, since <math>\mathbf{s}</math> denotes the new aberrisma steps added to the MOS) in the context of groundfault's [[aberrismic theory]], MOS substitution is intended to take advantage of extra potential symmetry when <math>a, c</math> or <math>b, c</math> is not a coprime pair and mildly generalize the congruence substitution procedure for building [[balanced]] words to obtain non-balanced but still more "even" scales with simple [[generator sequence]] expressions (in the sense of being binary, i.e. using only two distinct generators).
-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. This article denotes the resulting scale <math>\mathsf{MOS\_subst}(a, b, c; \mathbf{x}, \mathbf{y}; k),</math> where <math>\mathbf{y}</math> is the new step size inserted, <math>\mathbf{x}</math> is the step size in the starting MOS identified with <math>\mathbf{y}</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>).
+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. This article denotes the resulting scale <math>\mathsf{MOS\_subst}(a, b, c; \mathbf{y}, \mathbf{z}; k):</math>
+<math>\displaystyle{ \mathsf{MOS\_subst}(a, b, c; \mathbf{y}, \mathbf{z}; k) := \mathsf{subst}( a\mathbf{w}(b + c)\mathbf{X}[0|{a+b+c-1}] , \mathbf{X}, b\mathbf{y}c\mathbf{z}[k|b+c-k-1] ) }</math>
+where <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>).
 ==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.