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)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, 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 aL(b + c)X and (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)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, 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 aL(b + c)X and (b/(b, c))m (c/(b, c))s instead. | ||
== Worked-out example == | == Worked-out example == | ||
For 5L2m4s, we exploit gcd(b, c) = 2 and | 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). | ||
== Open questions == | == Open questions == | ||
# Is the length of the shortest guided generator sequence related to the length of the filler MOS? It could hold when the scale pattern has the divisibilities that this procedure is intended to take advantage of. | # Is the length of the shortest guided generator sequence related to the length of the filler MOS? It could hold when the scale pattern has the divisibilities that this procedure is intended to take advantage of. | ||
# How can we ensure that the deletion of s steps from a scale thus "aberrismized" recovers the MOS version of aLbm? | # How can we ensure that the deletion of s steps from a scale thus "aberrismized" recovers the MOS version of aLbm? | ||