Reply to Daniel Wolf

Daniel,

The notion of psychoacousitc "stability" that I eluded to is clearly only a
part of what constitutes "stability" in a musical context. I'd be happy to
use a different word if you could suggest a better one -- "rootedness?"
Nevertheless, it has some explanatory power, as I have shown.

The masking will of course be far worse in the subharmonic case, where the
combination tones will tend to fill up all of frequency space, while in the
harmonic case, the combination tones will be restricted to integer multiples
of the fundamental. For root-position triads in the same register, the
combination tones will be spaced 2.5 times more widely for major than for
minor; as the harmonic limit increases past 5, this number grows very
rapidly. If I misunderstood you here, could you give a concrete example of
what you were talking about?

>The intuitionist (like an algorithm maker, or a student of musical
>cognition) proceeds in strictly chronological steps. If from the unity 1:1,
>the twoity is observed, and from the sum of 1+1, 2 is constructed, yielding
>the first harmonic interval, only then may the first subharmonic (the
>inversion) be constructed.

What if you're dealing with string lengths instead of frequencies? Certainly
string lengths are a more "intuitive" quantity than unobservably fast
frequencies! (This is intended to be a reductio ad absurdum).

>Your examples of string lengths
>etc. I found curious, as the (pre-wave length) music theory using this
>instrumentality was inversionally flexible in the extreme, and ran counter
>to Pauls strong harmonic series approach.

Actually, looking at string lengths would lead you to believe that
subharmonic relationships are even simpler that harmonic ones. An a priori
numerological analysis will not favor harmonics or subharmonics in any way,
despite your claim that there are differences. It is only by looking at the
psychophysical phenomena that the inequivalency manifests itself.

>There is
>indeed a (mathematical and physical) boundary problem for the infinitesimal
>(with null or the vacuum) that the infinite does not have.

Please explain this in as technical terms as you like. I have a degree in
physics.

Taking your stamements at face value, why does this "mathematical boundary
problem" affect frequencies and not string lengths, oscillatory periods, or
wavelengths? If you were correct, then these three quantities would be
happier producing subharmonic series (in which these quantities march
unimpeded up to infinity) than harmonic series (in which they plummet down
to the depths of the infinitesimal). Again, note that I am being facetious
and my intent is to demonstrate the absurdity of your arguments.

-Paul E.


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Date: Thu, 3 Oct 1996 08:38:55 -0700
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Saturday, December 12th at 8pm
Bad Girrls Studios
209 Green St.
Jamaica Plain (Boston), MA

I will be performing with Mad Duxx, an improvisational space-music
ensemble featuring me on microtonal guitar and keyboards, "Chicken"
David Allen didjeridu, "Phonetic" Brett Barbaro on Roland groovebox, and
Gabriel Jones on saxophone, normal guitar, etc. There will be two
opening acts (including Gabriel's own music) and the evening will begin
with a "radio orchestra" jam for which all audience members are
encouraged to bring a portable radio. Admission is $5, $4 if you bring a
radio.

For more information, call me at (617) MAD-DUXX.