Geometry of Linear Least Squares Problems

This information comes from a book called Elementary Linear Algebra by
Anton and Rorres, 9th Edition, Applications Version, Not For Sale in
North America. Your favorite academic library should have a book with
similar material.

A linear least squares problem is defined on p.333 as finding x to minimize

||A x - b||

As a regular temperament, x is a column vector containing the
generator sizes, b is the weighted just intonation point (JIP) and A
is the mapping matrix, with vals as columns.

They then define W as the column space of A, which is to say any val
that belongs to the temperament class, including things like vals that
don't contain integers, or any tuning of the temperament class. We
already know about this. The orthogonal projection of b on W is

proj_W b = A x

So you project the temperament onto the JIP, and the result is the
tempered (and weighted) tuning map.

There's no direct geometric model for x, which somebody asked about a
while back, but the above might help you to find it. There's more
working up to p.335 where they give a formula for proj_W b that looks
alarmingly like the pseudoinverse, so it won't help to repeat it.

Graham