The Frobenius tuning

Given a real square matrix M, the Frobenius norm on M is defined as the
square root of the sum of the entries mij of M; in other words, it is:

||M||_Frob = sqrt(trace(transpose(M) * M)))

This is the easiest matrix norm to work with.

Given a regular temperament, the projections belonging to the
temperament can be defined as the square matricies which have a zero
eigenspace on the left, spanned by the kernel of the temperament, and
a 1 eigenspace on the right, spanned by the vals of the temperament.
The Frobenius tuning can be defined as the minimum such matrix M in
terms of Frobenius distance from the identity I; this is what I gave
an example of yesterday. We can also call the tuning map we get from
multiplying
M * JIP as the Frobenius tuning.

In the case where the temperament is an equal temperament, the
Frobenius tuning is extremely easy to compute. This is because the
matrix in question is symmetrical, which can be proven by a coordinate
change to coordinates consisting of a basis for the kernel, plus the
val (as a monzo.) The off-diagonal elements only appear on the top
row, and go to zero for the Frobenius tuning; transforming coordinates
back again gives a symmetrical matrix, since this is symmetrical.

The consequences boil down to this: the inverval whose exponents are
defined by the val for the temperament is invariant under the
Frobenius tuning. Hence, to find the Frobenius tuning for <12 19 28|,
we simply find that octave stretch which makes |12 19 28> invariant.
We want to find the value k such that

12 + 19p3 + 28p5 = k(12 + 19(19/12) + 28(28/12))

which is 0.9973..., leading to an octave of 1196.778 cents. This is
therefore the Frobenius tuning of 5-limit 12-et.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> In the case where the temperament is an equal temperament, the
> Frobenius tuning is extremely easy to compute.

Aside from being easy to compute, the Frobenius tuning for equal
temperaments has a stronger theoretical justification than I had
realized at first. The most theoretically significant matrix norms
depend on the singular values, which are the square roots of the
eigenvalues of the matrix times its transpose. For equal temperaments,
all but one of the singular values is zero, and so various matrix
norms all coelesce into giving the same answer as the Frobenius norm.
Hence, while the Frobenius tuning for other regular temperaments is
simply the easiest to compute among these various competing norm
tunings, for equal temperaments it is not only trivial to compute, it
is rather special.