MOS substitution

A procedure for obtaining a ternary scale with arbitrary aLbmcs scale signature, intended to take advantage of extra symmetry when a, c or b, c is not a coprime pair and generalizing the congruence substitution procedure of building balanced words to obtain more non-balanced but still more "even" scales.

Take for example d = gcd(a,c), let a' = a/d and c' = c/d. Consider the MOS word (a+c)Xbm. The most even arrangement of a'-many L steps and c'-many s steps is the MOS a'Lc's, so this method would prescribe following the latter MOS to fill in the X's. Fixing a choice of which X in (a + c)Xbm you start from, of which there are gcd(a + c, b) choices, you have to choose a mode of a'Lc's, of which there are lcm(a, c) choices. When a' = c' = 1, we obtain a balanced (equivalently MV3) 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 aL(b + c)X and (b/(b,c))m (c/(b,c))s instead.

For 5L2m4s, we obtain mLsLsLmLsLs, sLsLmLsLsLm, and sLmLsLsLmLs. The first two are a chiral pair that are billiard scales, and the last is achiral, but it is not deletion-MOS.