Hein ten Horn wrote:

Jim Kelley wrote:

On Jul 5, 9:38 pm, John Fields wrote:

Sure enough, I heard the beat even though it came from different
sources, but I couldn't quite get it down to DC even with the
scope's trace at 0V.

Of course you heard beats. What you didn't hear is the sum of the
frequencies. I've had the same setup on my bench for several months.
It's also one of the experiments the students do in the first year
physics labs. Someone had made the claim a while back that what we
hear is the 'average' of the two frequencies. Didn't make any sense
so I did the experiment. The results are as I have explained.


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.

gr, Hein

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.

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.

Regards,
jk