RE: [tuning-math] Re: geometric complexity
- From: "Paul H. Erlich" <PErlich@xxxxxxxxxxxxxxxxx>
- To: "'tuning-math@xxxxxxxxxxxxx'" <tuning-math@xxxxxxxxxxxxx>
- Date: Fri, 19 Jul 2002 00:55:22 -0400
>What's your definition of complexity in general?
Just about the same as yours, but . . .
>> 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?
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? And if you
don't obey this "odd" rule, your metric is not accurately reflecting the
relative discordance of the consonant intervals. This is one of the most
important things a metric should do.
But i won't cry about it for the purposes of a paper that only goes through
7-limit. 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)
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