Multi-Dimensionality

I wrote:

>... to me, prime,
> odd, and scale-interval-generator are three distinctly different and
equally
>valid kinds of dimensionality''

Daniel Wolf responded:

>I believe that definitions like these have a lot to do with what exactly
>you want
>your music theory to do.

True. But I find these definitions to arise from the properties of the
numbers themselves.

>If you are making a pre-compositional theory...

That's basically what I'm up to, yes. I want to better understand the
geometrical relationships between integer-ratio intervals, and how they occur
in various JI tunings, partly because I find them beautiful (abstractly and
sonically), but also so that I can better capture that beauty in pieces of
music.

>... the way musical materials are
>articulated is essential to their interpretation.

True. Culture and personal taste will always play the primary role. But
there is a beauty in mathematical order that tends to transcend culture and
taste. They say that laughter and mathematics are the two truly universal
languages. I think that's why music is sometimes considered to be one as
well. It's not always one, but it can be - as a vehicle for expressing human
emotion in a context of mathematical order.

As to the use of dimensionalities in music, a composer might choose to
consider only one kind and restrict himself with only its related limit-type.
That's fine. He hasn't done away with the other kinds, he's simply chosen
to downplay their ramifications.

On to the examples:

>(1) I compose a piece consisting only of chromatically planed (parallel)
>Major ninth
>chords without fifths in just intonation. If the orchestration is such th=
>at a listener can hear the
>individual voices of these chords, then 9 is being articulated
>independently of 3. Since
>9 is here relatively prime, how could 9 be here distinguished a 'real'
>prime?

I tried that. It may be cultural or the result of intermodulation
distortion, but my ear filled in the fifths. Is there such a concept as
"relative primality"? That's a new one on me. I don't know the higher maths
too well -- but I thought a number was either prime or it wasn't. Maybe
that's where my ignorance is leading me to a false conclusion?

>(2) I write a chorale in triads that goes I - IV - V - I. There is a 9:8
>relationship between IV and V chords, but this is usually weakly
>articulated and the V of V (3^2)
>relationship is heard as more important.

I'm not sure where that example is leading. There is no V of V there of the
type that I'm familiar with, which is a II chord acting as a dominant to a
following V chord.

>(3) I compose a sound installation consisting of standing sine waves using
>only prime
>multiples of a given fundamental and 9 times that fundamental. Depending
>upon where
>I am in the room, I will hear the 9 as either an independent entity or in
>connection with 3.

If 3 and 9 had a common sound source and were in phase with each other, no
place in the installation would cause 3 to phase-cancel without also
cancelling the 9, or constructively interfere with the 3 without also
boosting the 9. That could not be true of any two different primes. Seems
to me this one helps my argument.

If the idea is that the sines are tuned to 2, 3, 5, 7, 9, 11, 13, 17, 19, 23,
.., and the room has a mode at the frequency of the 1, then I believe you
would always hear the implied fundamental no matter where you stood in the
room, and the 9 would act to enhance that implied tone except where it (the
9) phase-cancelled.

>(4) Using a TX81Z (with a resolution of 1.56 cent deviations from 12tet),=
> I perform
>a melody based upon random walks over a lattice with two dimensions: 3s and
>9s. The
>best approximation of 9 does not coincide with the best approximation of
>3^2. For all practical purposes, 3 and 9, in this temperament, are
>relatively prime.

Matters of the TX tuning table aside, a tuning is conceivable in which
intervals of approximately 3/x and 9/x are used, and one or both differ from
their corresponding exact integer-ratio values greatly enough that the one
being called "3," when squared, yeilds an interval significantly different
from the one called "9" that the two ("3"^2 and "9") would function as
distinct entities in the scale or in harmonies. Is that the idea? In such a
case, IMHO, integers are inadequate to describe the nature of that tuning,
and to use "3" and "9" to refer to those intervals is to invite confusion.

I missed the relation between your fifth example and your point.

David



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