shortest path (mostly Erlich)

[Keenan]
>Some folk may think that 8:7 is no more dissonant than 7:6 (odd-limit would
>say so).

With odd limit both are 7-limit ratios.

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

How so? One dimension for every odd factor. Or as you say, the (odd limit
minus one) over two. Some ratios will appear twice, you just take the most
direct route, and you're golden.

[Erlich]
>But if you have an 11-axis in the lattice, then that means you're
>considering 11-limit intervals consonant, so you should also consider
9->limit intervals consonant, and you should have a 9-axis too. That would
make >9:5 a single step of length lg(9).

Exactly!

[Erlich]
>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

Only up to the point you've declared consonant! The problem with your
algorithm is it has no way to measure intervals outside of those you
declare consonant.

[Erlich]
>but since the competing harmonic complexity formulae didn't impose any
limit >to the number of dimensions in the lattice, why should mine?

It shouldn't, but it should have a way to consider the shortest-route
factorization of intervals above the declared odd limit.

[Erlich]
>Nah, I just lost you. What did you mean? I know that 5/4 is the space
>between 5/1 and 4/1. So?

So the mediant of 5/1 and 4/1 is 9/2. Hence (n+d)/2. Is this an incorrect
generalization for dyads?

[Erlich]
>>Why not just throw a 2-axis on the lattice and weight by log prime/odd
>>limit as before?
>
>Of course that would have no effect on the metric.

? I'm saying consider 2 a factor just like 3 or 5. This would be like
Hahn's whole-number-limit.

[Erlich]
>>And 11/6 does?
>
>I never claimed much importance for local minima of harmonic entropy,

That's true, you're on record for that (do I understand correctly that you
prefer to take a given value, and see the probabilities of ratios from
there?). But didn't this thread start by you saying something about the
lack of a local minima at 11/8?

Carl

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

>How so? One dimension for every odd factor. Or as you say, the (odd
limit
>minus one) over two.

Yes, but the odd limit would have to be specified in advance.

>Some ratios will appear twice, you just take the most
>direct route, and you're golden.

I thought I wrote that!

>>But if you have an 11-axis in the lattice, then that means you're
>>considering 11-limit intervals consonant, so you should also consider
>9->limit intervals consonant, and you should have a 9-axis too. That
would
>make >9:5 a single step of length lg(9).

>Exactly!

Yeah, this is where the Hahn/Op de Coul algorithm went awry.

>>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

>Only up to the point you've declared consonant! The problem with your
>algorithm is it has no way to measure intervals outside of those you
>declare consonant.

Well, if you are allowed to declare an odd limit for consonance, it
_can_ measure intervals that are composites of consonant intervals
(using the lattice), it just can't measure intervals using higher prime
numbers. But since those prime numbers aren't used in any consonances,
then they're completely analogous to irrationals in the prime-based
measures we've discussed -- irrevocably outside the system.

>>but since the competing harmonic complexity formulae didn't impose any
>limit >to the number of dimensions in the lattice, why should mine?

>It shouldn't, but it should have a way to consider the shortest-route
>factorization of intervals above the declared odd limit.

Yes, and Paul Hahn's algorithm, with vector entries for all odd numbers
up to the limit instead of just primes, looks like the right way to
implement my idea for that.

>>Nah, I just lost you. What did you mean? I know that 5/4 is the space
>>between 5/1 and 4/1. So?

>So the mediant of 5/1 and 4/1 is 9/2.

Right.

>Hence (n+d)/2. Is this an incorrect
>generalization for dyads?

I'm still not following you.

>>>Why not just throw a 2-axis on the lattice and weight by log
prime/odd
>>>limit as before?
>
>>Of course that would have no effect on the metric.

>? I'm saying consider 2 a factor just like 3 or 5. This would be like
>Hahn's whole-number-limit.

Well, if you weight by prime or odd limit like you said, then the 2 will
never come into the weighting of any intervals, except those which are
octave-equivalent to the unison. But if you use the whole-number-limit,
then you do have something worth considering.

>>>And 11/6 does?
>
>>I never claimed much importance for local minima of harmonic entropy,

>That's true, you're on record for that (do I understand correctly that
you
>prefer to take a given value, and see the probabilities of ratios from
>there?). But didn't this thread start by you saying something about
the
>lack of a local minima at 11/8?

Well, if there's no local minimum at 11/8, then there's really no ground
for distinguishing 11/8 from nearby rational or irrational intervals,
based on harmonic entropy. Of course, combination tones, and, for
timbres with strong 8th and 11th partials, beating and critical-band
roughness, may provide 11/8 with some "specialness".