lattice metrics and octaves

I think we should all be able to agree that 5/3 is more consonant than 8/5,
and that 5/4 is more consonant than 6/5. Which is a contradiction, with
octave equivalency...

[Erlich]
>If you make 5/4 simpler than 6/5, then you're making 8/5 simpler than
>5/3. Can you swallow that? Not me!

...as Erlich points out.

[Hahn]
>If I were to get picky about it, the reverse would probably be
true--leaving >in factors of 2 and going to whole-number-limit (instead of
odd-limit), 6 is >closer to 5 than 8, and that's how I'd rank them aurally.

If you mean 6/5, 5/4, and 8/5, this works for me. The n+d weighting puts
6/5 in the middle of 5/4 and 8/5. I'm not getting a feeling for which
(middle, or closer to 5/4) is more accurate.

I have a problem with limit weightings in general. Both odd and integer
limit give the 9/4 and 9/7 equal consonance, for example. For this reason,
I prefer the something like n+d.

n+d itself gets off the boat around the 9-limit, unfortunately. So
howabout log(n*d)? I tried it for 126/125...

lg(126*125) = 13.9430642 [just the fraction]

lg(2) + 2lg(3) + 3lg(5) + lg(7) = 13.943063 [prime factors 126*125]

2log(30) + log(35) = 14.9430642 [shortest lattice route 6/5, 6/5, 7/5]

...they're all the same (I used base2 logs). Can anybody tell if this
solves the problem concerning Paul Hahn's original algorithm? Of course,
you'd expect lg(n*d) and and the sum of the logs of the absolute values of
the prime factors to be the same. But if it's always the same as the
quickest lattice route, it makes calculating this metric rather easy.

[Erlich]
>If you can't swallow octave-equivalent formulations at all, you're missing
>out on something very useful, since it is often assumed that all
octave->equivalents of all scale tones will be used...

Why not just throw a 2-axis on the lattice and weight by log prime/odd
limit as before?

[Hahn]
>Hmm--Carl, can you swallow 5/4 = 5/3?

I think so. In fact, I can also accept the n+d weighting, giving the 5/3
as slightly more consonant.

Carl Lumma wrote,

>I have a problem with limit weightings in general. Both odd and
integer
>limit give the 9/4 and 9/7 equal consonance, for example. For this
reason,
>I prefer the something like n+d.

Again, odd limit only applies when you're considering all octave
equivalents together; 9/4 may be more consonant than 9/7, but 16/9 vs.
14/9? As I've said, I like n+d too if you're being octave-specific.

>n+d itself gets off the boat around the 9-limit, unfortunately.

How so?

>So
>howabout log(n*d)?

That doesn't get off the boat?

>I tried it for 126/125...

>lg(126*125) = 13.9430642 [just the fraction]

>lg(2) + 2lg(3) + 3lg(5) + lg(7) = 13.943063 [prime factors 126*125]

>2log(30) + log(35) = 14.9430642 [shortest lattice route 6/5, 6/5, 7/5]

>...they're all the same (I used base2 logs).

One looks 1 bigger than the other two.

>Can anybody tell if this
>solves the problem concerning Paul Hahn's original algorithm? Of
course,
>you'd expect lg(n*d) and and the sum of the logs of the absolute values
of
>the prime factors to be the same. But if it's always the same as the
>quickest lattice route, it makes calculating this metric rather easy.

I don't see how to visualize this metric.

>>If you can't swallow octave-equivalent formulations at all, you're
missing
>>out on something very useful, since it is often assumed that all
>octave->equivalents of all scale tones will be used...

>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. You probably meant
something slightly different.

On Tue, 16 Mar 1999, Carl Lumma wrote:
> I think we should all be able to agree that 5/3 is more consonant than 8/5,
> and that 5/4 is more consonant than 6/5. Which is a contradiction, with
> octave equivalency...

Which tells me that, given octave equivalence (which I think Paul E. and
I are both making as a simplifying assumption, realizing full well that
the model loses some accuracy because of it), it's simpler not to try to
make such fine distinctions, and just weight all 7-limit intervals the
same, all 5-limit intervals the same, etc.--which is what we were doing.

> Why not just throw a 2-axis on the lattice and weight by log prime/odd
> limit as before?

This would work, I _think_. Haven't worked out all the details yet, but
I think it would work pretty well.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "How about that? The guy can't run six balls,
-\-\-- o and they make him president."

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