Last answer to Paul

1) I have three degrees in music, so perhaps you can bear with a bit of the
history of theory. In western tonal theory, there are two major lines of
thought one, which we will call _voice leading theory_ and identified with
Fux, and continuing through Shenkerian and other methods, bases analyses
upon the existing chord structure, described from the sounding bass tone;
the other we can call _fundamental bass_ theory, identified with Rameau, is
based upon the discovery of roots that my or may not be present. In the
first model, rule systems are based primarily on melodic movement within
single voices and then between them in counterpoint, while the second model
is based upon a harmonic relationships. (Your example of a Rameau rule
about seconds in the Fundamental bass is not present in all versions of his
theories, which he changed periodically). In practice the division between
these models is not entirely straightforward, but comparing the work of a
voice leading composer like Brahms with a fundamental bass composer like
D-Indy or Faure is instructive. It is interesting that the both the Riemann
chord classification scheme - an essentially just intonation model, perhaps
familiar to those who know the work of Vogel - and Schoenbergs functional
harmony hedge their bets by making it clear that voice leading (via species
counterpoint) is prior to harmonic theory.

My attempt to locate your work within this theoretical framework is an
attempt to locate its utility for musical analysis.

2) In that only pairs of tones will share harmonic series in a subharmonic
chord, the masking will be at a lower amplitude than for a harmonic chord
where all tones share a single series.

3) A division of a string into aliquot parts (subharmonic) is finite.
Successive harmonic divisions are not. Example: the division of the string
into four equal parts yields four pitches of the subharmonic sequence 1/1,
4/3, 2/1, 4/1 and no more.

4) The period (for example, one second) is a function of our notation. If
you have an alternative notion please describe it.

5) Huygens was the founder of wave theory and arguably the first modern
physicist; his methods were truly counterintuitive for his time and remains
so. (I am not saying that he was wrong, just that the physics and the
cognition of music are different and depend upon different apprehensions -
hence my attraction to intutionist models where temporal perception is
prior to spatial, and computation is placed in temporal sequence).

Daniel Wolf

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