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Your hearing is fairly insensitive to frequency dependent phase shift, but
One of the interesting phenomena in hearing is that of turn-off (it can be
done in a gradual enough way that the turn-off edge doesn't introduce a
lot of high frequency components). A tone is present, but humans give no
indication of hearing it *until the tone is stopped*. Then they say, "Oh,
I WAS hearing something".
Hearing is tricky!
A question --
You claim (correctly) that ears have upper limits on what can be heard. I
have been told by people in music that although the ear cannot hear above,
say 20,000 Hz, the sound will be different if those higher frequencies are
there or not. Or, put another way, the trained ear can sense the higher
inaudible frequencies being present or not. Anyone have experience, or
knowledge in this area?
Are the "cross terms", "difference terms", etc. necessary to describe the
pressure fluctuations in the air, or do they apply only to the perception
of the sound?
I discussed a finite series, only, not an infinite series.
The problem is that in real instruments, nonlinearities are typically
present (there is argument that they make *all* the difference), and so
one hears "cross terms", "difference terms", and other bits which are
certainly not present in the signal originally used to drive the
instruments. The presence of these terms is, of course, dependent on the
volume at which the instruments are played, and that's not usually covered
by strictly modal decomposition.
As to your question about the possibility of an infinite series forming
the sum which is heard:
1) it's possible, so long as the sum is square-integrable (we have
to conserve energy, after all);
2) in principle, it doesn't matter, because our ears have upper
limits on what they can hear anyway.