thingmajig

>I don't see how to visualize this metric.

If PaulE doesn't get it, I wouldn't expect anybody to get it. Here's
another go...

I'm suggesting that each rung of the lattice be weighted by all its
factors, rather than just its highest one. Paul Hahn's original alogrithm
was...

1. take the prime factorization
2. sum negative and positive exponents seperately
3. take the larger of the absolute values of the sums

This is equivalent to taking the odd limit of each rung on the shortest
route to the target. But taking the logs of the prime factors as you do
the above procedure does *not* give the same result as taking the log of
the odd limit of each rung.

My algorithm...

1. take the prime factorization
2. take the absolute values of all the exponents
3. sum them all up

I think this is equivalent to taking the n*d of each rung on the simplest
path to the target. The question is, is taking the logs of the prime
factors during the above procedure equivalent to taking the log(n*d) of
each rung on the shortest path? For the one case I considered, it (almost)
was.

Carl

On Wed, 17 Mar 1999, Carl Lumma wrote:
> I'm suggesting that each rung of the lattice be weighted by all its
> factors, rather than just its highest one. Paul Hahn's original alogrithm
> was...
>
> 1. take the prime factorization
> 2. sum negative and positive exponents seperately
> 3. take the larger of the absolute values of the sums
>
[snip]
>
> My algorithm...
>
> 1. take the prime factorization
> 2. take the absolute values of all the exponents
> 3. sum them all up

The problem with this is, you end up with things like 9:8 being more
dissonant than 6:5. Unless you discard octave equivalence and leave in
all the 2s. Which, I realize, Carl _is_ advocating. But then it comes
out the same as lg(n*d) again, I'm pretty sure.

I'd like to point out, before we all get into another round of yelling
at each other, that this is all angels dancing on the heads of pins
without some idea of what we're using all these metrics _for_. For
example, I'm more interested in in developing a measure of what you
might call _conceptual_ dissonance, as opposed to _perceptual_
dissonance. The latter is, basically, roughness. Think of the former
more in terms of, "which is harder to sing (accurately), a 3:2 or a
7:5?"

--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."

NOTE: dehyphenate node to remove spamblock. <*>

On Wed, 17 Mar 1999, Paul Hahn wrote:
>> 1. take the prime factorization
>> 2. take the absolute values of all the exponents
>> 3. sum them all up
>
> The problem with this is, you end up with things like 9:8 being more
^^^^
> dissonant than 6:5.

Less, I meant less. Sorry.

--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."

NOTE: dehyphenate node to remove spamblock. <*>