MOS substitution: Difference between revisions

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The following holds for <math>S = \mathsf{mos\_subst\_aberrize}(a, b, L, c, k)</math> (resp. <math>\mathsf{mos\_subst\_aberrize}(a, b, m, c, k)</math>):

# If the template MOS <math>T = T(m,X) = M_{b,n}(m,X;n-1)</math> (resp. <math>T(L,X)=M_{a, b+c}(L,X;n-1)</math>) is primitive, let <math>r</math> the count of X steps in a chosen (reduced) generator of <math>T.</math> Since <math>r</math> must be coprime to <math>n</math>, <math>r</math>-steps in the filling MOS <math>F = M_{a,c}(L,s;k)</math> (resp. <math>M_{b,c}(m,s;k)</math>) come in exactly 2 sizes, <math>iL+js</math> and <math>(i-1)L+(j+1)s</math> (resp. <math>im+js</math> and <math>(i-1)m+(j+1)s</math>), and taking this generator of <math>T</math> results in a [[generator sequence]] of length <math>q</math>. Letting <math>\mathsf{GS}(g_1, ..., g_{q})</math> be this generator sequence, <math>g_j</math> is either <math>pm + iL + js</math> or <math>pm + (i-1)L + (j+1)s,</math> according as the size ''j''-th ''r''-step in the sequence of stacked <math>r</math>-steps in the chosen mode of <math>F</math> is <math>~~pm +~~ iL + js</math> or <math>~~pm +~~ (i-1)L + (j+1)s.</math> (We could have chosen to use the "darkest" mode of <math>T</math> instead, which corresponds to taking the circle of (''n − r'')-steps in ''F'' and is thus also valid.)

# If the template MOS <math>T = T(m,X) = M_{b,n}(m,X;n-1)</math> (resp. <math>T(L,X)=M_{a, b+c}(L,X;n-1)</math>) is primitive, let <math>r</math> the count of X steps in a chosen (reduced) generator of <math>T.</math> Since <math>r</math> must be coprime to <math>n</math>, <math>r</math>-steps in the filling MOS <math>F = M_{a,c}(L,s;k)</math> (resp. <math>M_{b,c}(m,s;k)</math>) come in exactly 2 sizes, <math>iL+js</math> and <math>(i-1)L+(j+1)s</math> (resp. <math>im+js</math> and <math>(i-1)m+(j+1)s</math>), and taking this generator of <math>T</math> results in a [[generator sequence]] of length <math>q</math>. Letting <math>\mathsf{GS}(g_1, ..., g_{q})</math> be this generator sequence, <math>g_j</math> is either <math>pm + iL + js</math> or <math>pm + (i-1)L + (j+1)s,</math> according as the size ''j''-th ''r''-step in the sequence of stacked <math>r</math>-steps in the chosen mode of <math>F</math> is <math>iL + js</math> or <math>(i-1)L + (j+1)s.</math> (We could have chosen to use the "darkest" mode of <math>T</math> instead, which corresponds to taking the circle of (''n − r'')-steps in ''F'' and is thus also valid.)

== Example ==

For 5L2m4s, we exploit gcd(b, c) = 2 and substitute 2m4s into the template MOS 5L6X (LXLXLXLXLXX). Since 2m4s has three distinct modes (ssmssm, smssms, and mssmss) and 5L6X is primitive, we obtain three distinct scales: LsLsLmLsLsm, LsLmLsLsLms, and LmLsLsLmLss. The first two are a chiral pair of billiard scales, and the last is achiral but not deletion-MOS. 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), notably representing all 3 possible rotations of (L+s, L+m, L+s).

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 The following holds for <math>S = \mathsf{mos\_subst\_aberrize}(a, b, L, c, k)</math> (resp. <math>\mathsf{mos\_subst\_aberrize}(a, b, m, c, k)</math>):
-# If the template MOS <math>T = T(m,X) = M_{b,n}(m,X;n-1)</math> (resp. <math>T(L,X)=M_{a, b+c}(L,X;n-1)</math>) is primitive, let <math>r</math> the count of X steps in a chosen (reduced) generator of <math>T.</math> Since <math>r</math> must be coprime to <math>n</math>, <math>r</math>-steps in the filling MOS <math>F = M_{a,c}(L,s;k)</math> (resp. <math>M_{b,c}(m,s;k)</math>) come in exactly 2 sizes, <math>iL+js</math> and <math>(i-1)L+(j+1)s</math> (resp. <math>im+js</math> and <math>(i-1)m+(j+1)s</math>), and taking this generator of <math>T</math> results in a [[generator sequence]] of length <math>q</math>. Letting <math>\mathsf{GS}(g_1, ..., g_{q})</math> be this generator sequence, <math>g_j</math> is either <math>pm + iL + js</math> or <math>pm + (i-1)L + (j+1)s,</math> according as the size ''j''-th ''r''-step in the sequence of stacked <math>r</math>-steps in the chosen mode of <math>F</math> is <math>pm + iL + js</math> or <math>pm + (i-1)L + (j+1)s.</math> (We could have chosen to use the "darkest" mode of <math>T</math> instead, which corresponds to taking the circle of (''n &minus; r'')-steps in ''F'' and is thus also valid.)
+# If the template MOS <math>T = T(m,X) = M_{b,n}(m,X;n-1)</math> (resp. <math>T(L,X)=M_{a, b+c}(L,X;n-1)</math>) is primitive, let <math>r</math> the count of X steps in a chosen (reduced) generator of <math>T.</math> Since <math>r</math> must be coprime to <math>n</math>, <math>r</math>-steps in the filling MOS <math>F = M_{a,c}(L,s;k)</math> (resp. <math>M_{b,c}(m,s;k)</math>) come in exactly 2 sizes, <math>iL+js</math> and <math>(i-1)L+(j+1)s</math> (resp. <math>im+js</math> and <math>(i-1)m+(j+1)s</math>), and taking this generator of <math>T</math> results in a [[generator sequence]] of length <math>q</math>. Letting <math>\mathsf{GS}(g_1, ..., g_{q})</math> be this generator sequence, <math>g_j</math> is either <math>pm + iL + js</math> or <math>pm + (i-1)L + (j+1)s,</math> according as the size ''j''-th ''r''-step in the sequence of stacked <math>r</math>-steps in the chosen mode of <math>F</math> is <math>iL + js</math> or <math>(i-1)L + (j+1)s.</math> (We could have chosen to use the "darkest" mode of <math>T</math> instead, which corresponds to taking the circle of (''n &minus; r'')-steps in ''F'' and is thus also valid.)
 == Example ==
 For 5L2m4s, we exploit gcd(b, c) = 2 and substitute 2m4s into the template MOS 5L6X (LXLXLXLXLXX). Since 2m4s has three distinct modes (ssmssm, smssms, and mssmss) and 5L6X is primitive, we obtain three distinct scales: LsLsLmLsLsm, LsLmLsLsLms, and LmLsLsLmLss. The first two are a chiral pair of billiard scales, and the last is achiral but not deletion-MOS. 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), notably representing all 3 possible rotations of (L+s, L+m, L+s).