MOS substitution: Difference between revisions

'''MOS substitution''' is a procedure for obtaining a ternary scale with arbitrary scale signature aLbmcs. Originally developed by Inthar for the purpose of adding aberrisma steps in an orderly manner to a MOS pattern aLbm (which we write in place of aLbs 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.

Take for example d = (a, c) (:= gcd(a, c)), let a' = a/d and c' = c/d. Consider the MOS word (a + c)Xbm, 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)Xbm 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 aL(b + c)X and filling MOS (b/(b, c))m (c/(b, c))s instead.  

We tentatively denote the resulting scale <math>\mathsf{mos\_subst\_aberrize}(a, b, x, c, k),</math> where <math>x \in \{L, m\}</math> is the step size identified with s by the template MOS and <math>k \in \{0, 1, ..., (b + c)/(b,c)-1\}</math> is the brightness of the mode of the filling MOS used (0 corresponds to the darkest mode).

== Facts ==

Let <math>M_{a,b}(x,y;k)</math> be the mode of axby that would have brightness k if x were L and y were s. For example, <math>M_{a,b}(5,2;5)(x,y) = xxyxxxy.</math> Let <math> n = a+b+c</math> and <math>q = a + c</math> (resp. <math>b + c</math>).

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> (resp. <math>pL + im + js</math> or <math>pL + (i-1)m + (j+1)s,</math>) according as the ''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> (resp. <math>im + js</math> or <math>(i-1)m + (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.)

# Suppose that the perfect generator of ''T'' that we use subtends r < |T|/2 steps. Assume without loss of generality that the template MOS identifies L and s steps. Suppose the perfect generator is tL + us and the imperfect generator is (t - 1)L + (u + 1)s. Then the interval class of (r + 1)-steps has either (a) tL + (u + 1)s and (t - 1)L + (u + 2)s, or (b) tL + (u + 1)s and (t + 1)L + us.

#* In case (b), S becomes a mos after deleting s steps for k = 0, ..., v, where v is the number of generators stacked to obtain (t + 1)L + us in the filling MOS F.

== 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).