Odd-Limit Lattices

I did RMS calculations for arbitrary sets of intervals here:

http://x31eq.com/composite.pdf

The simplest way of adapting that to odd-limits is to average over all
distinct intervals in the limit with only odd numbers. That is, for
the 5-limit, {1:1, 3:1, 5:1, 3:5, 1:3, 1:5, 5:3}. This implies a
scale stretch that considers only these intervals and not octave at
all. That's wrong but not so wrong that it won't give sensible
results, so let's try.

As it happens, the only intervals that matter are {3:1, 5:1, 3:5}.
You can verify the result with the whole thing if you like. The
un-normalized metric comes out as

(1 0 1)(1 0) = ( 2 -1)
(0 1 -1)(0 1) (-1 2)
(1 -1)

In the 7-limit, the metric is

(1 0 -1 0 0 -1)( 1 0 0) ( 3 -1 -1)
(0 1 1 0 -1 0)( 0 1 0) = (-1 3 -1)
(0 0 0 1 1 1)(-1 1 0) (-1 -1 3)
( 0 0 1)
( 0 -1 1)
(-1 0 1)

That is what Gene gives for the BCC lattice in

http://lumma.org/tuning/gws/sevlat.htm

I got it wrong in another message. You have to multiply out
(-x_1+x_2+x_3)^2 + (x_1-x_2+x_3)^2 +(x_1+x_2-x_3)^2.

The variance of three samples is

<x^2> - <x>^2 = (x_1^2 + x_2^2 + x_3^2)/3 - (x_1 + x_2 + x_3)^2/9

Because it'll be normalized later, you can multiply through by 9

(x_1^2 + x_2^2 + x_3^2)*3 - (x_1 + x_2 + x_3)^2
= 3 x_1^2 + 3 x_2^2 + 3 x_3^2 - x_1^2 - x^2_2 - x_3^2 - 2 x_1 x_2 - 2
x_2 x_3 - 2 x_2 x_3
= 2 x_1^2 + 2 x_2^2 + 2 x_3^2 - 2 x_1 x_2 - 2 x_2 x_3 - 2 x_2 x_3

The quadratic form is

( 2 -1 -1)
(-1 2 -1)
(-1 -1 2)

That doesn't match the above. However, what we're really looking at
is the octave-equivalent part of a 4x4 metric, where the octave
coefficients are set to zero. That's how my standard deviation of
errors works out in the octave-equivalent case. So the actual metric
is a mix between a standard deviation and an RMS.

What this tells us is:

1) Gene was looking at standard deviations first, but didn't call them such.

2) An odd-limit RMS will give an odd-limit lattice as its dual, up to
the 7-limit.

3) The standard deviation error looks like the correct
octave-equivalent, Tenney-weighted error.

Point (2) breaks down beyond the 7-limit because the primes don't have
equal weight. This is what Tenney weighting is supposed to fix. The
true 9-limit metric happens to be

(15, -3, -3)
(-3, 4, -1)
(-3, -1, 4)

You can approximate this by using the 7-limit metric but giving 3
twice as much weight as the other primes. That gives an overall
metric of

(12, -2, -2)
(-2, 3, -1)
(-2, -1, 3)

There's no prime weighting that gives the true 9-limit metric when
combined with the 7-limit metric.

Still, there is an octave-equivalent 9-limit tone-space lattice that
follows from the 9-limit RMS. And Gene did say how to find the dual
of a quadratic form -- it's the inverse of the matrix. So in this
case, we have, as integers in their lowest terms,

(5, 5, 5)
(5, 17, 8)
(5, 8, 17)

That must define the geometry of the natural 9-limit lattice. It
can't give 3:1 and 9:1 equal weight but it makes a reasonable
compromise. I don't know what the lattice looks like :P

Point (3) follows because the weighted primes always have equal weight
by definition. These matrices are the same as those for the STD error
as a function of the weighted octave-equivalent errors. I haven't
worked out the actual TOP-RMS type error you get by using such a
metric. It may not be a true limit to infinity of anything sensible
because the 9-limit matrix isn't of the right form. I wonder if there
is a standard form -- I couldn't work it out for the composite errors
PDF which is why I defined the variance-like function.

Graham