On Jul 5, 9:38 pm, John Fields wrote:
On Thu, 05 Jul 2007 18:37:21 -0700, Jim Kelley
wrote:

John Fields wrote:

You missed my point, which was that in a mixer (which the ear is,
since its amplitude response is nonlinear) as the two carriers
approach each other the difference frequency will go to zero and the
sum frequency will go to the second harmonic of either carrier,
making it largely appear to vanish into the fundamental.

Hi John -

Given two sources of pure sinusoidal tones whose individual amplitudes
are constant, is it your claim that you have heard the sum of the two
frequencies?

---
I think so.

So if you have for example, a 300 Hz signal and a 400 Hz signal, your
claim is that you also hear a 700 Hz signal? You'd better check
again. All you should hear is a 300 Hz signal and a 400 Hz signal.
The beat frequency is too high to be audible. (Note that if the beat
frequency was a separate, difference signal as you suggest, at this
frequency it would certainly be audible.)

A year or so ago I did some casual experiments with pure tones being
fed simultaneously into individual loudspeakers to which I listened,
and I recall that I heard tones which were higher pitched than
either of the lower-frequency signals. Subjective, I know, but
still...

Excessive cone excursion can produce significant 2nd harmonic
distortion. But at normal volume levels your ear does not create
sidebands, mixing products, or anything of the sort. It hears the
same thing that is shown on both the oscilloscope and on the spectrum
analyzer.

Interestingly, this afternoon I did the zero-beat thing with 1kHz
being fed to one loudspeaker and a variable frequency oscillator
being fed to a separate loudspeaker, with me as the detector.

My comments were based on my results in that experiment, common
knowledge, and professional musical and audio experience.

I also connected each oscillator to one channel of a Tektronix
2215A, inverted channel B, set the vertical amps to "ADD", and
adjusted the frequency of the VFO for near zero beat as shown on the
scope.

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.

jk

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

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

"Jim Kelley" wrote in message
...
....
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.

Ok.

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.

??

How do you come up with anything but a period of of the average of the two
for the enveloped waveform?

David L. Wilson wrote:

"Jim Kelley" wrote in message
...
...

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.


Ok.

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.


??

How do you come up with anything but a period of of the average of the two
for the enveloped waveform?

The error here is in assuming that the sin and cos terms in the
equivalent expression are representative of individual waves. They
are not. The resultant wave can only be accurately described as the
sum of the constituent waves sin(a) and sin(b), or as the function
2sin(.5(a+b))cos(.5(a-b)). That function, plotted against time
appears exactly as I have described. I have simply reported what is
readily observable.

jk

Jim Kelley wrote:

David L. Wilson wrote:

"Jim Kelley" wrote in message
...
...

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.


Ok.

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.


??

How do you come up with anything but a period of of the average of the
two for the enveloped waveform?

The error here is in assuming that the sin and cos terms in the
equivalent expression are representative of individual waves. They
are not. The resultant wave can only be accurately described as the
sum of the constituent waves sin(a) and sin(b), or as the function
2sin(.5(a+b))cos(.5(a-b)). That function, plotted against time
appears exactly as I have described. I have simply reported what is
readily observable.

jk


I would submit you plotted it wrong and/or misinterpreted the results.

craigm wrote:
Jim Kelley wrote:


David L. Wilson wrote:


"Jim Kelley" wrote in message
...
...


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.


Ok.

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.


??

How do you come up with anything but a period of of the average of the
two for the enveloped waveform?

The error here is in assuming that the sin and cos terms in the
equivalent expression are representative of individual waves. They
are not. The resultant wave can only be accurately described as the
sum of the constituent waves sin(a) and sin(b), or as the function
2sin(.5(a+b))cos(.5(a-b)). That function, plotted against time
appears exactly as I have described. I have simply reported what is
readily observable.

jk



I would submit you plotted it wrong and/or misinterpreted the results.

Always a possibility, admitedly. However the superposition of two
waves each having a different frequency does not yield a resultant
waveform having a constant period. But you are certainly welcome to
try to demonstrate otherwise.

jk

craigm wrote:
Jim Kelley wrote:
David L. Wilson wrote:
Jim Kelley wrote:

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.

How do you come up with anything but a period of of the average of the
two for the enveloped waveform?

The error here is in assuming that the sin and cos terms in the
equivalent expression are representative of individual waves. They
are not. The resultant wave can only be accurately described as the
sum of the constituent waves sin(a) and sin(b), or as the function
2sin(.5(a+b))cos(.5(a-b)). That function, plotted against time
appears exactly as I have described. I have simply reported what is
readily observable.

I would submit you plotted it wrong and/or misinterpreted the results.

Jim, if you'd like me to send you an Excel sheet about this,
please let me know.

gr, Hein

I've sent this post already once. For some strange reason it didn't
come up in rec.radio.shortwave (craigm?).
I only read rec.radio.shortwave these days.
(repost to: sci.electronics.basics, rec.radio.shortwave,
rec.radio.amateur.antenna, alt.cellular.cingular,
alt.internet.wireless)

Jim Kelley wrote:

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.



If you have two values, a and b, the average is (a+b)/2, which is precisely
the frequency in your above equation. So the sin(.5(a+b)) term is at the
average frequency.

The sin's term amplitude is modified by the cos term, 2cos(.5(a-b)). This
does not change the timing of the zero crossings of the sin term in any
way. Therefore the period of the resulting waveform is fixed.

The cos term does add a few additional zero crossings when it evaluates to
0, but there is no continuous variation in the period as you have
described.

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