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

"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.

Hein ten Horn wrote:

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.

The mathematics doesn't provide the possibility except, as I have
noted, for brief instants of time. There exists no "wave of average
frequency" in the frequency spectrum of the sum of two waves. A
Fourier analysis of the function doesn't reveal one. The ear doesn't
"produce" one. And I can tell you from personal and professional
experience that it does not hear one. (A triad chord would be truly
awful to experience if it did.)

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.

Yes. But when multiplied by a sinusoidal function of a different
frequency (as is the actual equation), the amplitude is affected in a
way which varies in both magnitude and sign with time, and which
affects both the peak spacing and the zero crossings differently from
one cycle to the next as a function of relative phase. If one defines
the period of a waveform as the length of one cycle of a waveform,
then this length of time varies in the way I have previously
described. Please consider using Mathematica or your favorite
plotting program to examine this for yourself.

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).

Apart from the mathematical support, that is also what I have found.
However, I believe this usage has been disappearing in recent years as
re-evaluation replaces reiteration as a means for producing text
books. All I can say is that it appears the claim may have been made
by someone without sufficient experience in the particular field. I
can find no support, anecdotal, phenomenological, psychoacoustical, or
mathematical for the contention (repeated by rote from what I can
tell) that the ear hears the average when the two frequencies are
arbitrarily 'close'.

I've never heard it, and I've been playing musical instruments for 47
years, doing audio electronics for almost 30, and physics for the last
20. The notion appears to me to be speculation based upon little more
than a perfunctory analysis of the underlying mathematics.

It might be more reasonable to claim that what is heard is a slight,
slow warble in frequency, back and forth, from one pitch to the other
accompanyied by a corresponding change in volume. But when the beat
frequency is low, the two pitches are so close together that the
difference between them is not discernable.

However, getting some empirical evidence should be a
rather easy piece of work.

Easier to say than do, certainly, but an interesting and enjoyable
endeavor nevertheless. :-)

jk

Jim Kelley wrote:

Hein ten Horn wrote:

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.

The mathematics doesn't provide the possibility except, as I have
noted, for brief instants of time. There exists no "wave of average
frequency" in the frequency spectrum of the sum of two waves. A
Fourier analysis of the function doesn't reveal one. The ear doesn't
"produce" one. And I can tell you from personal and professional
experience that it does not hear one. (A triad chord would be truly
awful to experience if it did.)

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.

Yes. But when multiplied by a sinusoidal function of a different
frequency (as is the actual equation), the amplitude is affected in a
way which varies in both magnitude and sign with time, and which
affects both the peak spacing and the zero crossings differently from
one cycle to the next as a function of relative phase.


How can the zero crossings be affected? Zero multiplied by any other value
is still 0. All zero crossings in sin( pi * (f_2 + f_1) * t ) occur at the
expected time. Multiplication by a cos term does not change a single one.
(It will add a few additional ones where the cos term evaluates to 0.)
There are no phase effects here.

If one defines
the period of a waveform as the length of one cycle of a waveform,
then this length of time varies in the way I have previously
described. Please consider using Mathematica or your favorite
plotting program to examine this for yourself.

Defining the period as time between zero crossings leads to the frequency
not changing as you describe.


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).

Apart from the mathematical support, that is also what I have found.
However, I believe this usage has been disappearing in recent years as
re-evaluation replaces reiteration as a means for producing text
books. All I can say is that it appears the claim may have been made
by someone without sufficient experience in the particular field. I
can find no support, anecdotal, phenomenological, psychoacoustical, or
mathematical for the contention (repeated by rote from what I can
tell) that the ear hears the average when the two frequencies are
arbitrarily 'close'.

I've never heard it, and I've been playing musical instruments for 47
years, doing audio electronics for almost 30, and physics for the last
20. The notion appears to me to be speculation based upon little more
than a perfunctory analysis of the underlying mathematics.

It might be more reasonable to claim that what is heard is a slight,
slow warble in frequency, back and forth, from one pitch to the other
accompanyied by a corresponding change in volume. But when the beat
frequency is low, the two pitches are so close together that the
difference between them is not discernable.

However, getting some empirical evidence should be a
rather easy piece of work.

Easier to say than do, certainly, but an interesting and enjoyable
endeavor nevertheless. :-)

jk