the chord-finding problem again

Heya Gene,

>I would work in 3-dimensions for the 9-limit, and just make 3
>half the size of 5 or 7. In other words,
>
>||3^a 5^b 7^c|| = sqrt(a^2 + 4b^2 + 4c^2 + 2ab + 2ac + 4bc)
>
>would be the length of 3^a 5^b 7^c. Everything in a radius of 2
>of anything will be consonant.

Any suggestions for how to find all integer solutions for
a,b,c... in something like this?

>If we like, we may adjust matters by multiplying through by the
>lcm of the exponents of the largest prime powers, so as to be
>able to work with integers.

Not sure if this is a start at answering the above question.

>This sort of thing is what I meant when I said I came upon
>hexanies and the like geometrically. This metric is useful partly
>because two octave equivalence classes separated by a distance of
>one or less are o-consonant, and by a distance of greater than
>one are o-dissonant.

1 or less? That's radius 1/2. Above you say radius 2!

>How's this as a method: using the standard o-limit metric, take
>everything in a radius of 1 of the unison, //

Now it's radius 1!

> // which should give you the o-limit diamond. Now take all
> subsets of size k, find the centroid by averaging the coordinates
>(which should be in the prime-power basis, so that in the 9-limit
>5/3 would be 9^(-1/2) * 5^1 * 7^0 = [-1/2, 1, 0], for instance)

9 isn't prime!

>and test if everything is within a radius of 1/2 of the centroid,
>in which case put it on your list. For larger values of o, this
>would be faster than simply testing for pairwise consonance.

Radius 1/2 again. I assume this metric works whether one starts
on a lattic note or not (I assume the centroid could be on a
point at which no pitch class is found, or even a point not
lying on a consonant dyad line)...

-Carl