Mossy thoughts

Let's see if I can shed some light on this murky mess.

Suppose (a, N)=1 (a is relatively prime to N) and we want a scale in
N-equal. We can define it as a*i mod N, where i is an element of an
index set S. Since a is invertible mod N, this defines all the
possible scales; the point of introducing a and S is that S can be a
set of contiguous numbers.

Now suppose we really wanted a scale in F*N-equal, consisting of F
transpositions of the same scale. Carl defines his version of this as

(a*i mod N) + N*j, for all 0 <= j < F, i in S. This is the same as
(a*i mod N) + N*j mod F*N.

I defined my version as

a*i + N*j mod F*N, for all 0 <= j < F, i in S

How are these related? a*i mod N and a*i mod F*N differ by a multiple
of N, so (a*i mod N) + N*j mod F*N is a*i + N*k mod F*N for some value
of k. If (N, F)=1 then if a*i + N*k = w mod F*N we can solve for k as
(w-a*i)/N mod F, and the two definitions are the same. I'll take a
look at what happens if N and F are not relatively prime.