MOS substitution: Difference between revisions

'''MOS substitution''' is a procedure for obtaining a ternary scale with arbitrary scale signature a'''L'''b'''m'''c'''s'''. Originally developed by Inthar for the purpose of adding aberrisma steps in an orderly manner to a MOS pattern a'''L'''b'''m''' (which we write in place of a'''L'''b'''s''' 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. (This article bolds steps '''L''', '''m''', '''s''', and '''X'''.)

<!-- Todo: Remove <math> tags -->

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.