More response to Paul

(1) Please read more carefully, my approach is independent of combination
tones, and is not affected one way or another by their presence. This
choice of independence - like the choice of just intonation as a mode - was
made to streamline matters, including the avoidance of pitch masking by
spectra of pitches belonging to a harmonic series. My use of subharmonic
characterization is strictly to identify a class of complexes that are more
_efficiently_ (I am momentarily at a loss for a more precise word) heard
not a part of a harmonic series. I know, for example, that I can hear,
identify, and reproduce the sine wave complex 500, 600, 750 Hz. I hear it
distinctly as a "minor" triad and neither as a mistuned harmonic series
segment where 500Hz = 2^n nor as a harmonic series segment over the
fundamental of 50Hz.

(2) Phase differences can have dramatic effects on pitch perception. With
long duration sound installations, the relative phase positions are
apparent in physical space. I recently heard an extraordinary installation
by Hauke Harder in Copenhagen, where phase relationships were essentially
the only dynamic element in the work. When phases were locked this entire
quality disappeared. Young has worked with _drifting_ phases and has made
some interesting psychoacoustical conjectures. (I have had similar - but
not precisely so - experiences in recording sine wave complexes from my
Rayna synthesize onto CD or DAT, where differences in sampling speed all
but destroyed entire works).

(3) A spacing theory would be a method of analysis and not a theory of
composition, although the information obtained might be useful to
composers. Last time I checked, composers were free to proceed with or
without a theory.

(4) I believe your procedure is closely tied to western musical materials
(I try to be a bit more global in my approach, but the simple matter is
that the number of repertoires with pitch complexes of three or more
members is limited; a parallel project of mine involves harmonization
procedures for repertoires with melodic properties distinctive from the
German folksongs upon which classical chorale harmonization is based); for
this reason, a coupling of your algorithm with an approach to chord
progression is worthwhile if not necessary, since your procedure demands
parameters only close to those offered by traditional western classical and
vernacular musics. In this (broadly defined) repertoire, contrast between
harmonic structures is a (if not THE) defining feature.

(5) For the Boomsliter and Creel example, I should have said harmonic
_progressions_, for the choice of 224/128 over 7/4 was aopparently made in
reference to position in a larger network of relationships, and not to a
stretching preference (although Linus Liu recently sent me an example of a
scale for Chinese music - which is primarily melodic - where a sequence of
just melodic intervals leads to an octave stretched by 81/80).

(6) I would appreciate it if some other readers would let me know if this
material is interesting or whether we should take the colloquy private; I
do not want this to burden anybody-s email facilities.

Daniel Wolf, Frankfurt

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