MOS substitution: Difference between revisions
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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 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. | ||
We tentatively denote the resulting scale <math>\mathsf{ | 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). | ||
== Facts == | == Facts == | ||
The following holds for <math>S = \mathsf{ | The following holds for <math>S = \mathsf{aberrize\_by\_mos\_subst}(a, b, c, \mathbf{L}, k)</math> (and after replacing '''L''' with '''m''' and a with b, for <math>\mathsf{aberrize\_by\_mos\_subst}(a, b, c, \mathbf{m}, k)</math> as well): | ||
Let <math>\mathsf{mos}(a,b;k)</math> be the mode of axby that would have brightness k if x were '''L''' and y were '''s'''. For example, <math>\mathsf{mos}(5,2;5)(x,y) = xxyxxxy.</math> Let <math> n = a+b+c</math> and <math>q = (a + c)/(a,c)</math>. | Let <math>\mathsf{mos}(a,b;k)</math> be the mode of axby that would have brightness k if x were '''L''' and y were '''s'''. For example, <math>\mathsf{mos}(5,2;5)(x,y) = xxyxxxy.</math> Let <math> n = a+b+c</math> and <math>q = (a + c)/(a,c)</math>. | ||
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==Examples== | ==Examples== | ||
=== 5L2m4s === | === 5L2m4s === | ||
To derive 5L2m4s as <math>\mathsf{ | To derive 5L2m4s as <math>\mathsf{aberrize\_by\_mos\_subst}(5, 2, 4, \mathbf{m}, k)</math>, we exploit gcd(b, c) = 2 and substitute 2'''m'''4'''s''' into the template MOS 5'''L'''6'''X''' ('''LXLXLXLXLXX'''). Since 2'''m'''4'''s''' has three distinct modes ('''ssmssm''', '''smssms''', and '''mssmss''') and 5'''L'''6'''X''' is primitive, we obtain three distinct scales: '''LsLsLmLsLsm''', '''LsLmLsLsLms''', and '''LmLsLsLmLss'''. All three scales admit short generator sequences of 2-steps, respectively GS('''L'''+'''s''', '''L'''+'''s''', '''L'''+'''m'''), GS('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s'''), and GS('''L'''+'''m''', '''L'''+'''s''', '''L'''+'''s'''), representing all 3 possible rotations of ('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s'''). | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ 5L2m4s as <math>\mathsf{ | |+ 5L2m4s as <math>\mathsf{aberrize\_by\_mos\_subst}(5, 2, 4, \mathbf{m}, k)</math> | ||
|- | |- | ||
!rowspan=2| ''k'' | !rowspan=2| ''k'' | ||
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=== 6L7m9s === | === 6L7m9s === | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ 6L7m9s as <math>\mathsf{ | |+ 6L7m9s as <math>\mathsf{aberrize\_by\_mos\_subst}(6, 7, 9, \mathbf{L}, k)</math> | ||
|- | |- | ||
!rowspan=2| ''k'' | !rowspan=2| ''k'' | ||