Group-theoretic generalizations of Fokker blocks, scales, epimorphicity

I've spent some time working out purely group-theoretic
generalizations of everything. This is long, so you may want to read
it in more than one sitting.

In entirely group-theoretic terms, for any free abelian group G,
particularly those whose elements are "surrogates" for musical
intervals:

==SCALE==
A "scale" is a function s: I -> G, such that I is a contiguous subset
of Z, and that s[0] = 0, the identity element in G.

For any scale, the "rank" of the scale is the rank of the smallest
subgroup of G containing Im(s).

For any n, s[n] - s[n-1] is called a "step" in the scale.

The "permutations" or "modes" m of the scale are each given by m[i-n]
= s[i]-s[n] for n in I; note that the elements i-n form a set which is
the translation of I in Z, and which is the domain of the permutation.
It can also be useful to classify the permutations of a scale
together; such a construction is called a "scale up to permutation,"
or when context makes clear, simply a "scale."

For any scale s, the "inversion" s* can be obtained by s*[i] =
-s*[-i], where the domain of s* is the reflection of I about the
origin in Z.

For any scale s, the "reversal" of the scale s' can be obtained by
s'[-i] = s[i].

==PERIODIC SCALE==
A "periodic scale" is a scale which has the property that there exists
an element p with s[q] = p, and for all integers n and k, s[n+kq] =
s[n]+kp. The element p such that q is non-negative and smallest is
called the period.

==WEAKLY EPIMORPHIC==
A scale s is "weakly epimorphic" if it has the property that there
exists an element h in Hom(G, Z) such that for all i in Z, h(s[i]) =
i; such an element is called the "organizer" for the scale. In
general, a subset S of G can be called "weakly epimorphic" if there
exists a weakly epimorphic scale with S in its image, though this
scale may not be unique.

==FOKKER ARENA==
A "Fokker arena" is defined as follows: pick an indivisible element e
in G, called the "equivalence element." Then, pick a set C of
rank(G)-1 indivisible elements, called "chromas," such that the set C
u {e} is linearly independent, and that the quotient group G/C is
torsion-free. The resulting tuple (e, C) is then referred to as a
Fokker arena.

To any arena are associated two indivisible elements in Hom(G,Z) which
are inverses of one another and send the elements in C to 0; these are
called the "organizers" for the arena, since they're organizers for
the Fokker blocks the arena generates, as defined below.

==FOKKER BLOCK==
As G also naturally has the structure of a free Z-module, we can
naturally embed it within the real vector space V which is the set of
all R-linear combinations of its elements, where G now forms a
lattice. Then, for any Fokker arena and associated equivalence
interval e, set of chromas C, and set of organizers O, the set of
associated "Fokker blocks" are those scales which can be constructed
as follows:

1) construct the set C u {0} + C u {0}, where + is the Minkowski sum;
note that the elements in this set are the vertices of a
parallelepiped in V.
2) translate this set by any vector such that the origin still lies
within its convex hull; call the resulting set C*.
3) obtain the convex hull Conv(C*), and calculate the Minkowski sum
Conv(C*) + {p | p = k*e for k \in R}, where e is the equivalence
interval of the Fokker arena.
4) obtain the intersection S of this set and the lattice which
corresponds to the embedded representation of G.
5) choose an o in the set of organizers O, and construct the function
f such that for all elements s in S, f(o(s)) = s.

Any function f obtainable in this way is called a Fokker block; note
that any such f also has the structure of a periodic scale. For any f,
the reversal can be obtained by simply choosing the other organizer in
step 5, which will also be a Fokker block. If we classify them
together we obtain a Fokker block up to reversal, which when context
makes clear may be referred to as a Fokker block with no additional
qualifier.

==DOMES==
For any arena, we can consider the set of Fokker blocks which are
equivalent up to permutation; these are the "domes" of the Fokker
block, either up to reversal or not.

==WAKALIXES==
Some Fokker blocks can be generated by starting from more than one
arena; these Fokker blocks are called wakalixes. For any Fokker block
which is a wakalix, all Fokker blocks sharing the same dome are also
wakalixes, so we may speak of domes themselves being wakalixes.

-Mike