Could I be permitted a mathematical comment?
There is no such thing as a linear
system (say a subwoofer) with an absolute upper frequency limit in
the actual world, only in mathematical theory.(A step function
of finite width is L2 and so it is the Fourier transform of something
else that is L2. But that is mathematics, not the real world).
The reason is that the ouput of such a system would not be able to start
and stop.
Namely, a signal the Fourier transform of which has its Fourier
transform =0 for all frequencies above some limit f cannot
be "compactly supported" in time. That is , it is not possible
that the signal =0 identically before some time and after some other
later time.
Of course, such mathematical models of the world are in practice
a matter of degree! But the mathematical principle does in part
explain why questions of this nature always sort of bog down.
Is bass directionally perceived? In practice, we almost all seem
to think so. But is this because of its higher frqeuency components?
The point I am trying to make is that there are ALWAYS higher frequency
components, except for the eternal "om" that started before time began and
that will continue into all eternity.(No offense I hope to believers
in the religious content here). Only that type of signal can be without
higher frequency components. Something that start and ends as nothing
in finite time cannot avoid having higher frequency
components--arbitrarily high in principle.
I have noticed over the years that psychoacoustics people hate to get
into experiments where by mathematical nature it is not very clear
what one is testing. This might be one reason why pyschoacoustics people
do not seem to have done all that much work on bass localization for
anything but sustained tones.
Incidentally, the fact that speakers in rooms produce impressions of bass
sounds coming from one place or another is not really a valid
determination of anything scientific to my mind. It mostly just shows
that if one likes the effect, having a bunch of subwoofers is a good idea.
Plus it illustrates the above principle.
Finally, my guess--and this is only a guess-- is that the sense of
localization of say a double bass player or contrabassoon player--and of
course they are
localized fine in practice--is almost entirely a matter of higher
frequency components in a simple minded sense, not the compact support
sense I just described, and not the time of arrival of the envelope of
the fundamental either.
I think I hear where the bass player in my
orchestra is(I am off to rehearsal in a minute) because of timing effects
on the attacks of the notes and so on. I would be amazed if much of
the rather precise localization that one perceives(one can surely point
straight at him with eyes closed, and also hear where each one is
separately, when we have two) is actually carried by envelope time- of-
arrival(whatever that might be interpreted to mean) of the fundamental
of the instrument. I think almost all of it is an aspect of attack and
texture and higher harmonics that involve much higher frequencies than
the 41 Hz of his/their bottom note(as I recall). There is a lot of this
energy at higher frequencies. (The double bass actually does not produce
all that much energy in the fundamental in fact. Much of what you
hear is harmonics).
But as I indicated , this is first of all only a guess, and in practice
the point is in the literal sense moot on account of the compact support
business. The word envelope is convenient, but the envelope being finite
in extent in time means that the sound contains in effect high frequency
components.
I shall crawl back under my mathematical rock now.
(The proof that an L2 function and its Fourier transform cannot both have
compact support was question 7 on the final exam in Fourier analysis
that I gave yesterday and that I was grading today so it was very much
on my mind!).
Robert
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