conanance and city-block

For measuring consonance, Erlich is right about the complex compound
intervals. They are almost certainly approximating other lower-numbered
ratios, and even if they weren't, the lattice probably wouldn't tell us
much about their consonance. However...

[Erlich]
>In that case, a formulation like mine treats them all at once.

Your formulation is just odd limit. And it doesn't make any sense
regarding city-block metrics. If you're going to go with city-block, you
must be interested in lattice-rung factorization. In which case you'll
probably want the rungs to represent consonances. Your version requires
rungs for every odd factor, consonant or not. Paul Hahn does agree that
odd-limit (not prime-limit) is a good measure of consonance, and he wants
to use it to gauge the complexity of other intervals that arise in music
(perhaps a bit like the prime-limit advocates do with prime numbers).

Carl

I wrote,

>>In that case, a formulation like mine treats them all at once.

>Your formulation is just odd limit. And it doesn't make any sense
>regarding city-block metrics.

It makes sense in the "sense" I explained. Odd limit is right if the
number of dimensions of the matrix is unlimited.

>If you're going to go with city-block, you
>must be interested in lattice-rung factorization. In which case you'll
>probably want the rungs to represent consonances.

Right, but how many dimensions would that require? The decision will
have a certain amount of arbitrariness.

>Your version requires
>rungs for every odd factor, consonant or not.

If you could decide how many dimensions you need, I would only have
rungs for consonant ones. That is, if you could decide the highest
odd-limit for which consonance applies, the number of dimensions would
be that minus one, over two (assuming you're using odd axes and
redundant lattice points, rather than prime axes and wormholes).

>Paul Hahn does agree that
>odd-limit (not prime-limit) is a good measure of consonance, and he
wants
>to use it to gauge the complexity of other intervals that arise in
music
>(perhaps a bit like the prime-limit advocates do with prime numbers).

So do I, but as Partch considered 9:5 a ratio of 9 and 11:5 a ratio of
11, and I agree with him, I had a problem with the Hahn/Op de Coul
algorithm, as it assigned the latter a smaller distance than the former.
Clearly 225 is way too high an odd number to be part of consonant ratios
(in lowest terms), but since the competing harmonic complexity formulae
didn't impose any limit to the number of dimensions in the lattice, why
should mine? I wanted to create something parallel so it could be used
by Manuel's program. Yes, it would be better to ask for an odd limit for
the lattice and compute the distance based on that. But the basic
insight is that a triangular lattice, with Tenney-like lengths, a
city-block metric, and odd axes or wormholes, agrees with the odd limit
perfectly, and so is the best octave-invariant lattice representation
(with associated metric) for anyone as Partchian as me.