Monzo question re Fokker blocks

/tuning/topicId_55446.html#55500

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> please do elaborate -- why only 3 blocks in this case?
> what's your general formula for finding how many blocks
> are "possible"?

Suppose we are in the p-limit, where p is the nth prime. Suppose also
we have a comma list {c1, ..., c_{n-1}} of n-1 intervals. Suppose this
list to be reduced (no torsion), so that if we wedge these together we
get an (n-1)-monzo whose complement is a val v0, which we may change
the sign of in order to make v0(2) = m a postive number. Since the
list is reduced, the gcd of the coordinates of v0 is 1; it is reduced.
We want to find all the Fokker blocks with the above comma list, which
will be blocks with m elements.

We choose an auxilliary interval c0, such that v(c0)=1. If we take the
nxn matrix with rows the monzos of {c0, c1, ..., c_{n-1}} then this
will be unimodular, ie with determinant +-1. We can invert it to get
an integral matrix, whose columns are vals, the first of which is v0,
so we may name them {v0, v1, ..., v_{n-1}}. Then for any rational
number q, we have that

q = c0^v0(q) c1^v1(q) ... c_{n-1}^v_{n-1}(q)

In particular, we can express 2 as

2 = c0^m c1^v1(q) ... c_{n-1}^v_{n-1}(2)

If ||x|| = floor(x+1/2) is a nearest-integer function, then for
offsets a0, a1, ..., an, -1/2 < ai < 1/2, we have that

1 = c0^||a0|| c1^||a1|| ... c_{n-1}^||a_{n-1}||

2 = c0^||m+a0|| c1^||m*(v1(2)/m)+a1|| ...
c_{n-1}^||m*(v_{n-1}(2)/m)+ a_{n-1}||

We can now find Fokker blocks, with these offsets, by setting the ith
block element as

Fokker[i] = c0^||i+a0|| c1^||i*(v1(2)/m)+a1|| ...
c_{n-1}^||i*(v_{n-1}(2)/m)+a_{n-1}||

If we look at where and how the a1 actually change anything, we find
we can restrict consideration simply to the a0=0, and ai to any
rational number of the form j/(2*m), with j odd and -m < j < m. This
gives us a finite number of things to check. We then end up with all
the various transpositions of the same scale; by reducing scales to a
standard form, we can cut the size of the list by a factor of m.