More on spacing: response to PAULE

(1) I cannot follow your logic regarding sine waves and the subharmonic.
Since the complexes we are discussing are within a finite section of the
audible frequency range, and we are probably not discussing ratios
involving harmonic partial larger than, say, 81, (although I have worked
with interesting - and perceivable - musical results through 128, and La
Monte Young through 512, partials), whether one thinks in terms of harmonic
or subharmonic is - initially - simply a matter of renotation. Thus, how is
it that you are able to decide that sine waves are "distinguished and
digestable" when described in harmonic context but suddenly not so when
renotated as subharmonic?

(2) Or do you define as subharmonic any complex without 2^n as the lowest
member? A spacing theory would compare the notations to determine which
characterization is more likely (if neither is, then the neutral
characterization is chosen). Your algorithm - and I do wish to examine it
in detail - strikes me as a root direction theory (and more in the
Hindemith than in the Rameau direction). The lack of register specificity
in your algorithm would however seem to be problematic in light of all
experimental evidence (and in light of traditional musics and
instrumentation). Both Clarence Barlow and James Tenney have profitably
included absolute frequency in their agorithms.

(3) You cite several psychoacoutistical certainties in support of your
algorithm, I am especially curious to learn of experimental results
supporting (a) "in a purely psychoacoustical sense, a chord played will
evoke certain interpretations in terms of the harmonic series"; this may be
what your algorithm tells me, and is a conjecture that I would like to
support, but I would like to find experimental evidence in this direction
(and I have encountered evidence to the contrary: see Boomsliter and Creel
on the 7th partial); (b) "the ear�s central pitch processor simply doesn�t
go that far"; I am completely confused by this: what do you mean by
"central pitch processor", and how far, exactly does it go? The harmonic
progression I presented, and its Pythagorean interpretation are simply too
obiquitous to be dismissed in this way.

(4) The sequential procedure would be something like the following: given
the complex 500 Hz,
600 Hz 750 Hz, the first ratio interpreted would be the 3/2 fifth of
750/500, the initial interpretation would be 3/2 in a harmonic series over
fundamental 250Hz; however, not finding an intermediate tone between 2 and
3, successive fifths in the series are scanned to find a similar complex,
examining each n(2,3) until an intermediate tone is located that matches
the complex. When, for example 10,12,15 over a fundamental of 50 Hz is
selected, then it is compared with its subharmonic notation /6,/5,/4. If
this characterization is more simple, then it is chosen. (Need I add the
fact that the difference tones of this complex are all within the spectrum
of 50 Hz? (The careful reader will note that I was cautious about the
register of this particular example)).

(5) All of the progressions that I described work - both acoustically, are
to found in existing repertoire, AND on paper. We (industrialized
westerners) have come, through historical reasons (largely inertia) to the
practice of using a single Pythagorean pitch notation to indicate a
diversity of wildly varying intonational environments. That certain tuning
systems, temperaments among them, may map onto harmonic series is probably
true, but it requires a definition of intervallic tolerance that seems to
me to be musically (i.e. culturally, and thus limited to particular times
and places) defined, and probably outside of the domain of psychoacoustics
proper: (a) tolerance may be a function of existing instrumental and
room-acoustic capacities; (b) tolerance may depend upon durational
conventions within a musical repertoire; (c) tolerance is a property of
warp-able analog recording and digital sampling rates; (d) La Monte Young
has demonstrated the perceptibility of ratios involving larger primes
through the use of extended duration sound environments).

(6) Your statement _the ear doesn�t continue its harmonic series math from
one chord to the next_ is unsupported, and probably unsupportable. It is
certainly musically untenable, unless you mean that the mechanism for
relating pitches in diachronic sequences to one another is independent from
the synchronic mechanism. This is possible, but seems inefficient in light
of the evidence that so much of acoustical processing is done by sharing
facilities. If you intend a separate _Gestalt_-type mechanism for chords
and another mechanism for successions, what do you do about broken chords?

(7) I cannot resist a few more remarks on the subharmonic. First, although
for musical purposes we never need get past fairly small numbers, while
every harmonic complex may be renotated as subharmonic and vice versa, due
to one of those nice paradoxes in the fundamentals of mathematics - and
beyond my ability to explain - the number of members in the harmonic series
is greater than that in the subharmonic series. Second, the _descent to the
infinitesimal_ represents to me a series of increasingly complex - indeed,
counterintuitive - calculations, hence my speculation about increased
mental activity. The ascription of quantum effects to the infinitesimal
comes from the intuitionist C. Hennix, resident mathematician at the
Institute for Psychoanalysis in Paris; Third, musical examples of the
subharmonic are readily available in the repertoire, cf Martin Vogel�s book
on the _Tristan_ chord, or this little example from the Rondo K.494, m
164-165, where the left hand sustains for two measures the chord c4 d4 f4
ab4, which moves (does not resolve) to c4 f4 a4, i.e subharmonic seventh
chord from c moving to f major 6-4. Mozart uses this chord, here and
elsewhere, in a voice leading distinctive from his use of diminished and
dominant seventh chords. By my spacing characterization, the fourth c-f
would be too important to overlook, and the traditional (in American
practice) _half diminished_ description seems not very useful. If this
were in just intonation, I would choose a subharmonic characterization /8
/7 /6 /5 over the harmonic 105 120 140 175. I am curious to learn how
Pauls algorithm will interpret this chord (16 18 21 26? or 24 27 32 38?).

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