Rank-3 scale theorems: Difference between revisions

Line 24:

A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if S, when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k. (Closest approx in what sense?)

In a mos (MV2), this follows from the fact that every k-step is of the form pL qs or (p+1)L (q-1)s, where p/q <= a/b <= (p+1)/(q-1). This in turn follows from the fact that a mos has a generator. If both the perfect gen and the imperfect gen had #L/#s > b/a, this would imply that n-1 perfect gens + 1 imperfect gen would not close at ~~the octave~~.

In a mos (MV2), this follows from the fact that every k-step is of the form pL qs or (p+1)L (q-1)s, where p/q <= a/b <= (p+1)/(q-1). This in turn follows from the fact that a mos has a generator. If both the perfect gen and the imperfect gen had #L/#s > b/a, this would imply that n-1 perfect gens + 1 imperfect gen would not close at a period multiple.

==== MV3 Theorem 1 ====

@@ Line 24: / Line 24: @@
 A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if S, when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k. (Closest approx in what sense?)
-In a mos (MV2), this follows from the fact that every k-step is of the form pL qs or (p+1)L (q-1)s, where p/q <= a/b <= (p+1)/(q-1). This in turn follows from the fact that a mos has a generator. If both the perfect gen and the imperfect gen had #L/#s > b/a, this would imply that n-1 perfect gens + 1 imperfect gen would not close at the octave.
+In a mos (MV2), this follows from the fact that every k-step is of the form pL qs or (p+1)L (q-1)s, where p/q <= a/b <= (p+1)/(q-1). This in turn follows from the fact that a mos has a generator. If both the perfect gen and the imperfect gen had #L/#s > b/a, this would imply that n-1 perfect gens + 1 imperfect gen would not close at a period multiple.
 ==== MV3 Theorem 1 ====