Jim Kelley wrote:
Hein ten Horn wrote:

We hear the average of two frequencies if both frequencies
are indistinguishably close, say with a difference of some few
hertz. For example, the combination of a 220 Hz signal and
a 224 Hz signal with the same amplitude will be perceived as
a 4 Hz beat of a 222 Hz tone.

I have also read this accounting, but from what I've been able to determine
it lacks mathematical and phenomenological support. Here's why. As two
audio frequencies are moved closer and closer together, there is no point
where an average of the two frequencies can be perceived. There is however
a point where no difference in the two frequencies is perceived. Obviously
if we cannot discern the difference between 220Hz and 224Hz (as an example),
we are not going to be able to discern half their difference either. I
suspect the notion may have originated from a trigonometric identity which
has what could be interpreted as an average term in it.

sin(a) + sin(b) = 2sin(.5(a+b))cos(.5(a-b))

A plot of the function reveals that cos(.5(a-b)) describes the envelope.
The period of the 'enveloped' waveform (or the arcane, beat modulated
waveform) then can be seen to vary continuously and repetitiously over
time - from 1/a at one limit to 1/b at the other. At a particular instant in
time the period does in fact equal the average of the two. But this is true
only for an instant every 1/(a-b) seconds.

The math is perfectly describing what is happening in the
course of time at an arbitrary location in the air or in the
medium inside the cochlea. Concerning the varying
amplitude it does a good job.
But does someone (here) actually know how our hearing
system interprets both indistinguishable(!) frequencies (or
even a within a small range rapidly varying frequency) and
how the resulting 'signal' is translated into what we call the
perception? Evidently the math given above doesn't
reckon with any hearing mechanism at all. Hence it cannot
rule out perceiving an average frequency.

For the rest I don't get your point on a varying period.
From a mathematical point of view the function

sin( pi * (f_2 + f_1) * t )

has a constant frequency of (f_2 + f_1)/2
and a constant period of 2/(f_2 + f_1).
This frequency is indeed the arithmetical average and
it is not affected by a multiplication of the function by
a relatively slow varying amplitude.

An interesting related experiment can be performed by setting a sweep
generator to sweep over a narrow range of frequencies. The range can be
adjusted as well as the sweep time. One can then study what sorts of
effects are discernible.

I have found that it is very difficult to fool the ear in some of the ways
that have been suggested. It does not appear, for example, that the claim
for 'perceiving the average' is valid for two arbitrarily close frequencies
any more than it is for any two other frequencies. But I would appreciate
learning of any contradictory research that you might be able to cite.

Apart from the mathematical support, I saw the average
frequency mentioned in several books on physics, unfortunately
without further enclosed proof (as far as I remember).
However, getting some empirical evidence should be a
rather easy piece of work.

gr, Hein