interval qualities; prime/odd

> But what about steely coldness? Is it there when the
> power is zero? Does a unison contain all potential
> interval qualities, to a greater degree than the intervals
> themselves? It would seem hard to argue that way.

This is a very intriguing way of posing the question.

My response would be that an exponent of 0 for both
numbers in the ratio of the interval (for any prime bases),
which will always give a unison, seems to have a unique
quality not shared by any other exponent-number.

Perhaps the fact that negative exponents give the complementary
intervals of their complementary positive exponents, and
therefore the same effect/ affect, lends support to this idea.

For example (I am describing intervals here, that is, dyads):

3^1 is equivalent to 3/2
3^-1 " 4/3

3^1 * 5^-1 " 6/5
3^-1 * 5^1 " 5/3

I don't think anyone would dispute that these pairs of intervals
exhibit exactly similar effect/ affect in a symmetrical way.
So perhaps a zero exponent (= 1/1), because it has no
complement, exhibits either a different property or no property
at all in this regard.



> 2 is the only prime that's even, but is that really all that
> important? 3 is the only prime that's a multiple of 3, and
> 5 is the only prime that's a multiple of 5.

I think it _is_ important. My statement about 2 having
special properties because it's the only even prime was
in response to the debate over whether the important
limits in harmony are the prime numbers or simply the
_odd_ numbers.

Any even-numbered harmonic identity
is assumed to represent the odd number to which it
can be reduced by dividing by 2. If 2 is divided by 2,
it's plain to see that it represents 1, which, however,
is not a prime, but in musical mathematics, is the _identity_,
the number which, when multiplied by any other number,
gives a product which is always that number itself.
This makes it obvious to me that 2 has special properties
in music which are possessed by no other number,
prime or composite-odd.

Joseph L. Monzo
monz@juno.com

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