Multi-Dimensionality

Daniel Wolf wrote in digest #1126:

>... Why should I have to speak
>in terms of four dimensions when I am only using three? Or, in the other
>direction, I might choose to treat nine as a separate dimension from three,
>in which case, Partch's diamond is viewed as six-dimensional rather than
>the five yielded by counting primes.

I can see the treatment of 9 as a separate dimension from 3 as legitimate.
But I think it's a different kind of dimensionality, if you will, from the
prime type. I can't fully explain this concept yet, but to me, prime, odd,
and scale-interval-generator are three distinctly different and equally valid
kinds of dimensionality. As Graham hinted, a case might even be made for
distinguishing between power-of-prime dimensionality and other odd-composite
("rhubarb"-number) dimensionality. In that case, one might choose on
mathematical principle, not merely whim, to allow 15, 21, 33, etc., in
ratios, but exclude 9, 25, 27, etc.

I lean toward the idea that prime dimensionality is the most basic kind
because it seems that there is an increasing level of difficulty for the ear
in accepting intervals involving greater primes, that is more intense than
the difficulty in acceptance of powers of a prime that has already been
accepted. So, for example, there may be a basic sense in which, in a context
where 3s are already prevalent but 5s are not, 3*3*3/16 is a little easier
for the ear to digest than 5/3, even though 27>15. Of course, that doesn't
mean you would tend to prefer 27/16, just that 5/3 might feel like it's
leaving home base in a different direction than the listener has experienced
up to that point.

Perhaps this idea of inverse proportionality of prime numbers to aural
digestibility (if it holds water) lies at the root of the apparent phenomenon
of octave invariance/equivalence. It may be that powers of 2 are so easily
accepted by the ear that it tends to see octave-separated pitches as the same
scale member. It also might be connected to the fact that, in addition to
its being the lowest prime factor, 2 is the only even prime.

I suggest that limits within each of the various forms of dimensionality be
used cooperatively: say, 5 prime-limit with a 27 odd-limit; or 13 prime-limit
with a 105 odd-limit. One could also impose power-limits: say, nothing
beyond cubes allowed (except of 2s to achieve the appropriate octave). One
might apply his odd limit only to "rhubarb" numbers while allowing pure
powers-of-primes to exceed it, or to have their own limit.

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