MOS substitution: Difference between revisions

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

'''MOS substitution''' is a procedure for obtaining a ternary scale with arbitrary scale signature <math>a\mathbf{L}b\mathbf{m}c\mathbf{s}</math>. 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 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 generalize the congruence substitution procedure for building [[balanced]] words to obtain non-balanced but still more "even" scales and simple generator sequence expressions (in the sense of using only two distinct generators) for them.

Note: This article bolds steps ~~'''~~L~~'''~~, ~~'''~~m~~'''~~, ~~'''~~s~~'''~~, ~~and '''X'''~~.

(Note: This article bolds steps <math>\mathbf{L}, \mathbf{m}, \mathbf{s}, \mathbf{x}.</math> For integers <math>m, n, \ (m, n) := \gcd(m, n).</math>)

Take for example d = (a, c) ~~(:= gcd(a, c))~~, let a' = a/d ~~and~~ c' = c/d. Consider the MOS word (a + c)'''X'''b'''m''', which we call the ''template MOS''. The most even arrangement of a'-many '''L''' steps and c'-many '''s''' steps is the MOS a'Lc's, so this method prescribes following the latter MOS, called the ''filling MOS'', to fill in the '''X''''s. Fixing a choice of which '''X''' in (a + c)'''X'''b'''m''' you start from, you have to choose a mode of a'Lc's. (Todo: count the distinct choices.) If a' = c' = 1 (equivalently if a = c), we obtain a balanced (thus MV3) ternary scale; when in addition b is odd, the scale is also SV3 and chiral, and we recover the two chiralities from the two modes of a'Lc's. Of course, one may do this using template MOS a'''L'''(b + c)'''X''' and filling MOS (b/(b, c))'''m''' (c/(b, c))'''s''' instead.

Take for example <math>d = (a, c)</math>, let <math>a^\prime = a/d, c^\prime = c/d.</math> Consider the MOS word (a + c)'''X'''b'''m''', which we call the ''template MOS''. The most even arrangement of a'-many '''L''' steps and c'-many '''s''' steps is the MOS a'Lc's, so this method prescribes following the latter MOS, called the ''filling MOS'', to fill in the '''X''''s. Fixing a choice of which '''X''' in (a + c)'''X'''b'''m''' you start from, you have to choose a mode of a'Lc's. (Todo: count the distinct choices.) If a' = c' = 1 (equivalently if a = c), we obtain a balanced (thus MV3) ternary scale; when in addition b is odd, the scale is also SV3 and chiral, and we recover the two chiralities from the two modes of a'Lc's. Of course, one may do this using template MOS a'''L'''(b + c)'''X''' and filling MOS (b/(b, c))'''m''' (c/(b, c))'''s''' instead.

We tentatively denote the resulting scale <math>\mathsf{aberrize\_by\_mos\_subst}(a, b, c, x, k),</math> where <math>x \in \{\mathbf{L}, \mathbf{m}\}</math> is the step size identified with '''s''' by the template MOS and k is the brightness of the mode of the filling MOS used (0 corresponds to the darkest mode, since '''L''' (or '''m''') > '''s''').

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-'''MOS substitution''' is a procedure for obtaining a ternary scale with arbitrary scale signature a'''L'''b'''m'''c'''s'''. Originally developed by Inthar for the purpose of adding aberrisma steps in an orderly manner to a MOS pattern a'''L'''b'''m''' (which we write in place of a'''L'''b'''s''' for convenience's sake, since '''s''' denotes the new steps added to the MOS) in the context of groundfault's aberrismic theory, MOS substitution is intended to take advantage of extra symmetry when a, c or b, c is not a coprime pair and generalize the congruence substitution procedure for building [[balanced]] words to obtain non-balanced but still more "even" scales and simple generator sequence expressions (in the sense of using only two distinct generators) for them.
+'''MOS substitution''' is a procedure for obtaining a ternary scale with arbitrary scale signature <math>a\mathbf{L}b\mathbf{m}c\mathbf{s}</math>. 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 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 generalize the congruence substitution procedure for building [[balanced]] words to obtain non-balanced but still more "even" scales and simple generator sequence expressions (in the sense of using only two distinct generators) for them.
-Note: This article bolds steps '''L''', '''m''', '''s''', and '''X'''.
+(Note: This article bolds steps <math>\mathbf{L}, \mathbf{m}, \mathbf{s}, \mathbf{x}.</math> For integers <math>m, n, \ (m, n) := \gcd(m, n).</math>)
 <!-- Todo: Remove <math> tags -->
-Take for example d = (a, c) (:= gcd(a, c)), let a' = a/d and c' = c/d. Consider the MOS word (a + c)'''X'''b'''m''', which we call the ''template MOS''. The most even arrangement of a'-many '''L''' steps and c'-many '''s''' steps is the MOS a'<b>L</b>c'<b>s</b>, so this method prescribes following the latter MOS, called the ''filling MOS'', to fill in the '''X''''s. Fixing a choice of which '''X''' in (a + c)'''X'''b'''m''' you start from, you have to choose a mode of a'<b>L</b>c'<b>s</b>. (Todo: count the distinct choices.) If a' = c' = 1 (equivalently if a = c), we obtain a balanced (thus MV3) ternary scale; when in addition b is odd, the scale is also SV3 and chiral, and we recover the two chiralities from the two modes of a'<b>L</b>c'<b>s</b>. Of course, one may do this using template MOS a'''L'''(b + c)'''X''' and filling MOS (b/(b, c))'''m''' (c/(b, c))'''s''' instead.
+Take for example <math>d = (a, c)</math>, let <math>a^\prime = a/d, c^\prime = c/d.</math> Consider the MOS word (a + c)'''X'''b'''m''', which we call the ''template MOS''. The most even arrangement of a'-many '''L''' steps and c'-many '''s''' steps is the MOS a'<b>L</b>c'<b>s</b>, so this method prescribes following the latter MOS, called the ''filling MOS'', to fill in the '''X''''s. Fixing a choice of which '''X''' in (a + c)'''X'''b'''m''' you start from, you have to choose a mode of a'<b>L</b>c'<b>s</b>. (Todo: count the distinct choices.) If a' = c' = 1 (equivalently if a = c), we obtain a balanced (thus MV3) ternary scale; when in addition b is odd, the scale is also SV3 and chiral, and we recover the two chiralities from the two modes of a'<b>L</b>c'<b>s</b>. Of course, one may do this using template MOS a'''L'''(b + c)'''X''' and filling MOS (b/(b, c))'''m''' (c/(b, c))'''s''' instead.
 We tentatively denote the resulting scale <math>\mathsf{aberrize\_by\_mos\_subst}(a, b, c, x, k),</math> where <math>x \in \{\mathbf{L}, \mathbf{m}\}</math> is the step size identified with '''s''' by the template MOS and k is the brightness of the mode of the filling MOS used (0 corresponds to the darkest mode, since '''L''' (or '''m''') > '''s''').