groups, matrices, etc.

Graham Breed comments on a possible approach to a general phonemics of
scales:

> I don't know the specifics of either group or category theory, but
> this is definitely something like what I do with matrices.

I'm no math whiz but matrices are a very classical way to represent groups
and are also a very nice example of a category. With a group you'd want the
matrices to all be square, that way any matrix can be multiplied by any other
matrix, the product is again square, etc. With categories one can allow
non-square matrices, and then you have to restrict multiplication so the rows
of one matrix match the columns of the other - i.e. the natural structure of
matrices lines up perfectly with the kind of restriction that the more
abstract category theory supports.

Just taking a finite subset of a group leaves you with, well, a finite subset
of a group. I don't know a general way to handle this with any elegance.

I'm just looking at this little book "Categories and Groupoids" by Philip
Higgins from 1971. It's pretty serious math, but it's where I got the idea of
just treating categories as algebraic objects in their own right. Usually
they're used more as a way to talk about the structure of other mathematical
objects. Anyway, since intervals are invertible, the natural phonemic
structure of scales might be a groupoid.

The best place to start is surely group theory. A fourth plus a tone is a
fifth. A scale is first of all some rule about how intervals combine. Then
one takes a subset of all the different possible combinations. The tricky
part is, is there any structure to how the subset is picked. The whole
groupoid tack is one blind grope toward some such structure.

Jim



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