Hein ten Horn wrote:

Hein ten Horn wrote:

quote
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.
unquote
(..)


From the example: there's no 222 Hz tone in the air.


That one I'd like to take back.
Obviously the superposition didn't cross my mind.
The matter is actually vibrating at the frequency
of 222 Hz. Not at 220 Hz or 224 Hz.

gr, Hein

You were correct before. It might be correct to say that matter is
vibrating at an average, or effective frequency of 222 Hz. But the
only sine waves present in the air are vibrating at 220 Hz and 224 Hz.
Obviously. It's a very simple matter to verify this by experiment.
You really ought to perform it (as I just did) before posting
further on the subject.

jk

Jim Kelley wrote:
Hein ten Horn wrote:
Hein ten Horn wrote:

quote
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.
unquote

From the example: there's no 222 Hz tone in the air.

That one I'd like to take back.
Obviously the superposition didn't cross my mind.
The matter is actually vibrating at the frequency
of 222 Hz. Not at 220 Hz or 224 Hz.

You were correct before.

That's a misunderstanding.
A vibrating element here (such as a cubic micrometre
of matter) experiences different changing forces. Yet
the element cannot follow all of them at the same time.
As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.
The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

It might be correct to say that matter is vibrating at an
average, or effective frequency of 222 Hz.

No, it is correct. A particle cannot follow two different
harmonic oscillations (220 Hz and 224 Hz) at the same
time.

But the only sine waves present in the air are vibrating
at 220 Hz and 224 Hz.

If so, we have a very interesting question...
What is waving here? A vacuum?
But don't take the trouble to answer.
You'd better distinguish the behaviour of nature and the
way we try to understand and describe all things.
As long as both sound sources are vibrating there are
no sine waves (220 Hz, 224 Hz) present, yet we do
use them to find the frequency of 222 Hz (and the
displacement of a vibrating element at a particular
location in space on a particular point in time).

Obviously. It's a very simple matter to verify this by experiment.

Indeed, it is. But watch out for misinterpretations of
the measuring results! For example, if a spectrum
analyzer, being fed with the 222 Hz signal, shows
that the signal can be composed from a 220 Hz and
a 224 Hz signal, then that won't mean the matter is
actually vibrating at those frequencies.

You really ought to perform it (as I just did) before
posting further on the subject.

I did happen to see interference of waterwaves
including some beautiful (changing) hyperbolic structures,
but no sign of any sine wave at all. So, with your kind
permission, here's my posting. ;-)

gr, Hein

"Hein ten Horn" wrote in message
...
Jim Kelley wrote:
Hein ten Horn wrote:
Hein ten Horn wrote:

quote
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.
unquote

From the example: there's no 222 Hz tone in the air.

That one I'd like to take back.
Obviously the superposition didn't cross my mind.
The matter is actually vibrating at the frequency
of 222 Hz. Not at 220 Hz or 224 Hz.

You were correct before.

That's a misunderstanding.
A vibrating element here (such as a cubic micrometre
of matter) experiences different changing forces. Yet
the element cannot follow all of them at the same time.

It does. Not identically but it does follow all of them.

As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.
The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

Your idea of frequency is informal and leaves out
essential aspects of how physical systems work.

You have looked at a segment of the waveform
and judged "frequency" based on a few peaks.
Your method is incomplete and cannot be applied
generally.

snip

Ron Baker, Pluralitas! wrote:
Hein ten Horn wrote:
A vibrating element here (such as a cubic micrometre
of matter) experiences different changing forces. Yet
the element cannot follow all of them at the same time.

It does. Not identically but it does follow all of them.

Impossible. Remember, we're talking about sound. Mechanical
forces only. Suppose you're driving, just going round the corner.
From the outside a fistful of forces is working on your body,
downwards, upwards, sidewards. It is absolutely impossible
that your body's centre of gravity is following different forces in
different directions at the same time. Only the resulting force is
changing your movement (according to Newton's second law).

As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.
The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

Your idea of frequency is informal and leaves out
essential aspects of how physical systems work.

Nonsense. Mechanical oscillations are fully determined by
forces acting on the vibrating mass. Both mass and resulting force
determine the frequency. It's just a matter of applying the laws of physics.


Question

Is our auditory system in some way acting like a spectrum analyser?
(Is it able to distinguish the composing frequencies from a vibration?)

Ron?
Somebody else?

Thanks

gr, Hein

On Jul 14, 12:42 pm, "Hein ten Horn"
wrote:
Ron Baker, Pluralitas! wrote:
Hein ten Horn wrote:
A vibrating element here (such as a cubic micrometre
of matter) experiences different changing forces. Yet
the element cannot follow all of them at the same time.

It does. Not identically but it does follow all of them.

Impossible. Remember, we're talking about sound. Mechanical
forces only. Suppose you're driving, just going round the corner.
From the outside a fistful of forces is working on your body,
downwards, upwards, sidewards. It is absolutely impossible
that your body's centre of gravity is following different forces in
different directions at the same time. Only the resulting force is
changing your movement (according to Newton's second law).

As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.
The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

Your idea of frequency is informal and leaves out
essential aspects of how physical systems work.

Nonsense. Mechanical oscillations are fully determined by
forces acting on the vibrating mass. Both mass and resulting force
determine the frequency. It's just a matter of applying the laws of physics.

Question

Is our auditory system in some way acting like a spectrum analyser?
(Is it able to distinguish the composing frequencies from a vibration?)

Ron?
Somebody else?

Thanks

gr, Hein

On Sat, 14 Jul 2007 19:51:40 -0000, Jim Kelley
wrote:

On Jul 14, 12:42 pm, "Hein ten Horn"
wrote:
Ron Baker, Pluralitas! wrote:
Hein ten Horn wrote:
A vibrating element here (such as a cubic micrometre
of matter) experiences different changing forces. Yet
the element cannot follow all of them at the same time.

It does. Not identically but it does follow all of them.

Impossible. Remember, we're talking about sound. Mechanical
forces only. Suppose you're driving, just going round the corner.
From the outside a fistful of forces is working on your body,
downwards, upwards, sidewards. It is absolutely impossible
that your body's centre of gravity is following different forces in
different directions at the same time. Only the resulting force is
changing your movement (according to Newton's second law).

As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.
The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

Your idea of frequency is informal and leaves out
essential aspects of how physical systems work.

Nonsense. Mechanical oscillations are fully determined by
forces acting on the vibrating mass. Both mass and resulting force
determine the frequency. It's just a matter of applying the laws of physics.

Question

Is our auditory system in some way acting like a spectrum analyser?
(Is it able to distinguish the composing frequencies from a vibration?)

---
Yes, of course.

The cilia in the cochlea are different lengths and, consequently,
"tuned" to different frequencies to which they respond by undulating
and sending electrical signals to the brain when the nerves to which
they're connected fire. See:

http://en.wikipedia.org/wiki/Organ_of_Corti



--
JF

"Hein ten Horn" wrote in message
...
Ron Baker, Pluralitas! wrote:
Hein ten Horn wrote:
A vibrating element here (such as a cubic micrometre
of matter) experiences different changing forces. Yet
the element cannot follow all of them at the same time.

It does. Not identically but it does follow all of them.

Impossible. Remember, we're talking about sound. Mechanical
forces only. Suppose you're driving, just going round the corner.
From the outside a fistful of forces is working on your body,
downwards, upwards, sidewards. It is absolutely impossible
that your body's centre of gravity is following different forces in
different directions at the same time. Only the resulting force is
changing your movement (according to Newton's second law).

As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.
The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

Your idea of frequency is informal and leaves out
essential aspects of how physical systems work.

Nonsense. Mechanical oscillations are fully determined by
forces acting on the vibrating mass. Both mass and resulting force
determine the frequency. It's just a matter of applying the laws of
physics.

Let me call you an idiot now and get that out of the way.
You're an idiot.
You don't know the laws of physics or how to
apply them.

How do you determine "the frequency"?
Show me the math.

What is "the frequency" of
cos(2pi 200 t) + cos(2pi 210 t) + cos(2pi 1200 t) + cos(2pi 1207 t)



Question

Is our auditory system in some way acting like a spectrum analyser?
(Is it able to distinguish the composing frequencies from a vibration?)

Ron?
Somebody else?

Thanks

gr, Hein

Ron Baker, Pluralitas! wrote:
Hein ten Horn wrote:
Ron Baker, Pluralitas! wrote:
Hein ten Horn wrote:

As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.
The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

Your idea of frequency is informal and leaves out
essential aspects of how physical systems work.

Nonsense. Mechanical oscillations are fully determined by
forces acting on the vibrating mass. Both mass and resulting force
determine the frequency. It's just a matter of applying the laws of
physics.

You don't know the laws of physics or how to apply them.

I'm not understood. So, back to basics.
Take a simple harmonic oscillation of a mass m, then
x(t) = A*sin(2*pi*f*t)
v(t) = d(x(t))/dt = 2*pi*f*A*cos(2*pi*f*t)
a(t) = d(v(t))/dt = -(2*pi*f)^2*A*sin(2*pi*f*t)
hence
a(t) = -(2*pi*f)^2*x(t)
and, applying Newton's second law,
Fres(t) = -m*(2*pi*f)^2*x(t)
or
f = ( -Fres(t) / m / x(t) )^0.5 / (2pi).

So my statements above, in which we have
a relatively slow varying amplitude (4 Hz),
are fundamentally spoken valid.
Calling someone an idiot is a weak scientific argument.
Hard words break no bones, yet deflate creditability.

gr, Hein

Hein ten Horn wrote:

That's a misunderstanding.
A vibrating element here (such as a cubic micrometre
of matter) experiences different changing forces. Yet
the element cannot follow all of them at the same time.
As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.

The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

Under the stated conditions there is no sine wave oscillating at 222
Hz. The wave has a complex shape and contains spectral components at
two distinct frequencies (neither of which is 222Hz).

It might be correct to say that matter is vibrating at an
average, or effective frequency of 222 Hz.


No, it is correct. A particle cannot follow two different
harmonic oscillations (220 Hz and 224 Hz) at the same
time.

The particle also does not average the two frequencies. The waveform
which results from the sum of two pure sine waves is not a pure sine
wave, and therefore cannot be accurately described at any single
frequency.

Obviously. It's a very simple matter to verify this by experiment.


Indeed, it is. But watch out for misinterpretations of
the measuring results! For example, if a spectrum
analyzer, being fed with the 222 Hz signal, shows
that the signal can be composed from a 220 Hz and
a 224 Hz signal, then that won't mean the matter is
actually vibrating at those frequencies.

:-) Matter would move in the same way the sound pressure wave does,
the amplitude of which is easily plotted versus time using
Mathematica, Mathcad, Sigma Plot, and even Excel. I think you should
still give that a try.

jk

Jim Kelley wrote
Hein ten Horn wrote:

That's a misunderstanding.
A vibrating element here (such as a cubic micrometre
of matter) experiences different changing forces. Yet
the element cannot follow all of them at the same time.
As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.

The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

Under the stated conditions there is no sine wave oscillating at 222 Hz.
The wave has a complex shape and contains spectral components at two
distinct frequencies (neither of which is 222Hz).

Not a pure sine oscillation (rather than wave), but a near sine
oscillation at an exact period of 1/222 s. The closer the source
frequenties, the better the sine fits a pure sine. Thus if you
wish to get a sufficient near harmonic oscillation, conditions like
"slow changing envelope" are essential.

It might be correct to say that matter is vibrating at an
average, or effective frequency of 222 Hz.


No, it is correct. A particle cannot follow two different
harmonic oscillations (220 Hz and 224 Hz) at the same
time.

The particle also does not average the two frequencies.

Hmm, let's examine this.
From the two composing oscillations you get the overall
displacement:
y(t) = sin(2 pi 220 t) + sin(2 pi 224 t)
From the points of intersection of y(t) at the time-axes you
can find the period of the function, so examine when y(t) = 0.
sin(2 pi 220 t) + sin(2 pi 224 t) = 0
(..)
(Assuming you can do the math.)
(..)
The solutions a
t = k/(220+224) with k = 0, 1, 2, 3, etc.
so the time between two successive intersections is
Dt = 1/(220+224) s.
With two intersections per period, the period is
twice as large, thus
T = 2/(220+224) s,
hence the frequency is
f = (220+224)/2 = 222 Hz,
which is the arithmetic average of both composing
frequencies.

The waveform which results from the sum of two pure sine waves is not a pure
sine wave, and therefore cannot be accurately described at any single
frequency.

As seen above, the particle oscillates (or vibrates) at 222 Hz.
Since the oscillation is non-harmonic (not a pure sine),
it needs several harmonic oscillations (frequencies,
here 220 Hz and 224 Hz) to compose the oscillation at 222 Hz.

Obviously. It's a very simple matter to verify this by experiment.

Indeed, it is. But watch out for misinterpretations of
the measuring results! For example, if a spectrum
analyzer, being fed with the 222 Hz signal, shows
that the signal can be composed from a 220 Hz and
a 224 Hz signal, then that won't mean the matter is
actually vibrating at those frequencies.

:-) Matter would move in the same way the sound pressure wave does,

To be precise, this is nonsense, but I suspect you're trying
to state somewhat else, and since I'm not able to read your
mind today, I skip that part.

the amplitude of which is easily plotted versus time using Mathematica,
Mathcad, Sigma Plot, and even Excel. I think you should still give that a
try.

No peculiarities found.

gr, Hein