Re: geometric complexity

On Thu, 18 Jul 2002 18:45:41 -0400 "Paul H. Erlich"
<PErlich@Acadian-Asset.com> writes:

> Anyway, why aren't we closing in on finality for the project? What
> exactly
> is Euclidean geometric complexity going to mean to a musician that
> our
> previous measures don't capture well?

I want a measure which applies to all temperaments, not just linear ones.

On Fri, 19 Jul 2002 00:36:20 -0400 "Paul H. Erlich"
<PErlich@Acadian-Asset.com> writes:
>
> First of all, i don't know where you're getting just linear ones
> from.

What's your definition of complexity in general?

> Secondly, i don't see what there is about a Euclidean, as opposed to
> a
> triangular-taxicab, metric that is going to be reflective of how we
> hear. In
> fact, it would seem especially important at the 9-limit and above to
> deviate
> from Euclid.

I was proposing using a Euclidean metric which did not give the same size
to all prime numbers; prime p would have length ln(p), and if p and q are
odd primes, with q>p, then
length p/q = length q/p = ln(q). This uniquely determines a Euclidean
metric.

>What's your definition of complexity in general?

>Just about the same as yours, but . . .

What specifically?

>> Secondly, i don't see what there is about a Euclidean, as opposed to
>> a
>> triangular-taxicab, metric that is going to be reflective of how we
>> hear. In
>> fact, it would seem especially important at the 9-limit and above to
>> deviate
> from Euclid.

>>I was proposing using a Euclidean metric which did not give the same
size
>>to all prime numbers; prime p would have length ln(p), and if p and q
are
>>odd primes, with q>p, then
>>length p/q = length q/p = ln(q). This uniquely determines a Euclidean
>>metric.

>Right, but first of all, do we or don't we have octave equivalence?

We do; this is a metric on octave classes.

>Secondly, the metric (if you replace "prime" with "odd") is inconsistent
for
>intervals like 9/5, right? You can't form a Euclidean figure for the
9-limit
>pentad such that all the intervals obey this "odd" rule, can you?

It doesn't treat 9 quite like a prime, but I don't think it does badly.
In this case we have

L(7/5) = ln 7 = 1.946
L(9/5) = sqrt(2 ln(3)^2 + ln(5)^2) = 2.237
L(11/5) = ln(11) = 2.398

The value 2.237 instead of ln(9) = 2.197 doesn't seem that horrible to
me.

On Fri, 19 Jul 2002 00:55:22 -0400 "Paul H. Erlich"
<PErlich@Acadian-Asset.com> writes:

Just redo the 5-limit and see how everyone feels about the
> rankings, and off we go . . . (but i'll keep harping on the question
> of a
> more elegant metric)

Which 5-limit temperaments do you regard as essential? How would you rate
128/125, 135/128, 250/243, 78732/78125,
393216/390625, 3125/3072 or 648/625? How about the funky systems such as
25/24, 27/25, 16/15, 10/9, 9/8? Where do you draw that line?