[tuning-math] Re: geometric complexity
- From: Gene W Smith <genewardsmith@xxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Sat, 20 Jul 2002 19:04:01 -0700
>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.
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