RE: Pitch spacings

Daniel-

Firstly, let me note that your assumption that "we are listening to an
orchestra of sine waves" does not bode well for the perception of
subharmonic complexes as such, since, as you state, the subharmonic complex
is recognized "by guiding tone (the lowest pitch
common to the spectra of all pitches in the complex)." I think your
observation that superparticulars are easier to tune relies on the existence
of difference tones, which actually beat against the virtual pitch in the
case of superparticulars (tuning is easy: just eliminate this beating).
However, I think you can modify the statement to say "an orchestra of
generic harmonic timbres played as a reasonably loud volume" without harming
your arguments.

Secondly, I find myself agreeing with much of your approach, especially
since I have included an algorithm for "evaluation of relative complexity,"
exactly as you defined it, as an appendix of a paper that will appear in
Xenharmonikon. I'm relying on Goldstein's 1973 JASA paper, along with some
number theory, for this algorithm. I can provide you with details of the
algorithm if you like, but let me note that I have found that for listeners
with slightly above-average pitch resolution, the 12-tET minor triad will be
heard mainly as 10:12:15 and as 16:19:24, the winner depending on voicing,
and assuming the triad is in a register that puts some audible partials in
Goldstein's optimal frequency range. The latter assumption my algorithm
cannot do without, so it is only good for finding the "worst-case" or most
complex representations of a particular tuning's intervals and chords. If
your algorithm succesfully extends this to the case of arbitrary given
register, I will congratulate you. (I suggest you take a look at Goldstein's
paper if you haven't yet).

I would not be surprised if you are correct that 4:5:6 is happier than
10:12:15 since it requires less brain effort. However, I don't think it
makes much musical difference whether the minor triad is heard as 10:12:15,
or 16:19:24, or a combination of the two. My algorithm actually gives a
greater salience to the p5 alone than to the minor triad, suggesting that
the minor triad is actually heard most significantly as 2:x:3, where x is a
non-harmonic tone not integrated into the virtual pitch sensation evoked by
the 2:3. (Unfortunately, I have no justification for directly comparing a
2-note salience to a 3-note salience, but it seems to work fine.) The
salient musical properties of the minor triad relative to the major triad
are: (1) the p5 defining a clear root, which is not, however, reinforced by
the third, and (2) a similar level of roughness and beating as is found in
the major triad. The sadness of the minor triad is then due simply to its
greater harmonic indefiniteness, without any extra roughness that could
contribute "anger" or "fear" into the perception.

-Paul


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