source file: mills3.txt


Subject:  Re: Multi-Dimensionality

From:  DFinnamore@aol.com

Let me start by reiterating that I find it perfectly acceptable to consider 9
or other odd numbers as a separate harmonic dimension, thereby validating the
usefulness of odd-number limits.  What I'm maintaining is that there is also
a valid prime dimensionality in which primes^x and composite numbers are
included in the dimensions of the prime numbers of which they are composed,
and that prime limits can be used instead of, or in conjunction with, odd
limits.

Daniel Wolf writes:

>Two numbers sharing no prime factors other than 1 are relatively prime.

in response to my response to:

>... then 9 is being articulated
>independently of 3. Since
>9 is here relatively prime, how could 9 be here distinguished a 'real'
>prime?

3 and 9 certainly do share a prime factor, unless, as in your 4th example,
"3" and "9" are really only rough approximations of 3 and 9, in which case
one or both should be given a more specific designation to clarify their
geometrical relationships.

>The V chord is in a V of V relationship to the IV chord.

OK.  If the IV were I, then the I would be V, and the V would be the V of V.
 So we pretend that the IV is a I and analyze the V of the I in terms of the
IV instead of in terms of the I.  Why?  Forgive my denseness - How does this
mean that 9 should not be considered part of the prime dimension on 3?

>You will encounter wave fronts for 9 alone in thirds of the distance
>between fronts shared by 3 and 9.

I think I'm getting your point of view here.  It's like the previous example.
 If the sine wave tuned to the 3rd overtone were temporarily considered the
fundamental of its own tone, then the wave that is the 9 of the fundamental
of which the 3 is the 3 would be the 3 of it.  That's just a really hard way
of saying that 3*3=9.  The fact is, if several other primes are present, they
cannot all cancel at once (unless they are very carefully manipulated to do
so via complete and separate control over the phase of each sine) and when
some of them combine with the 9, it will serve to reinforce the implied 1,
whether or not the 3 is cancelled.

>a real example where the particular tuning is at a
>relatively low level in the identity of a piece of music and that certain
>relationships - which may be paralleled by mathematical ideals - are not
>necessarily projected materially, but are understood by listeners to be
>present through cultural or compositional means.

I'm sure that is often the case.  Virtually nobody in the country music
business here in Nashville understands why I, IV, and V are the primary
chords - they just use 'em 'cause it's the traditional country thang to do.
 But anyone who has studied JI and American culture knows why country writers
wear out those chords - there are clear mathematical and cultural reasons.
 Yer average country listener would not likely appreciate the projection of
prime dimensionality even if it were made through a country-style recording.
 However that may be, it doesn't mean that purely prime dimensionality cannot
be projected materially through any composition.

David



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