"Hein ten Horn" wrote in message
...
| 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

Actually the human ear can detect a beat note down to a few cycles.