Re: The Overtone series


"Jack Campin - bogus address" <bogus@xxxxxxxxxxxxxxxx> wrote in message
news:bogus-550951.08334024052007@xxxxxxxxxxxxxxxxxxxxxx
the ear hears things in overtone series. I take it as fact. All
sound must go through the ear and therefore sound must "go through"
the overtone series. Music uses sound... and therefore music must
"go through" the overtone series.
You are assuming that Fourier analysis is always applicable.
Actually it is... fourier analysis is a mathematical device and it
has been proven that all functions that have certain properties can
be analyzed with fourier analysis. All known signals are, for all
practical purposes, known to be these types of functions(essentially
L^2 integrable functions or functions with finite energy).

If you are drawing the definition that widely you are certainly NOT
giving the overtone series any role at all. Fourier transforms
vastly generalize Fourier series, they are not at all the same thing.



What the hell are you talking that they are not the same thing? The fourier
transform is generalized from the case of a finite periodic function that of
an infinite case. No one said they were the same thing... But just cause
they are not exactly the same does not mean they are not related.

If these things were in no way related or even remotely related then fourier
analysis would be useless. DFT and CFT would not in any way be consistant...
yet they are. Take the DFT of a signal that is bandlimited and the fourier
transformer of it and the spectrum coincides.

I think you don't know really much about the math behind the fourier
transform.

http://en.wikipedia.org/wiki/List_of_Fourier-related_transforms

from wiki:
a.. If it is discrete, the Fourier transform becomes a DTFT.
b.. If it is periodic, the Fourier transform becomes a Fourier series.
c.. If it is both, the Fourier transform becomes


Now you really believe that if you take the fourier transform of a discrete
signal that you get a different answer?

Ultimately they are all the same thing... just cause there are slight
differences due to the circumstances doesn't mean anything.

Doesn't matter whats going on.... there is a transform that works.

The fact is not that its linear because nonlinearities can be
approximated
by linear functionals.... which is what Helmholtz does. He essentially
derives a partial differential equation of the physical model of a
membrane
and then shows that it creates new frequencies... precisely what is done
by
non-linearities(which you cannot create by a linear differential
equation)... These new frequencies come about by the sums and differenes
of
the original frequencies.

The overtones of a membrane don't fit the overtone series. See Helmholtz,
page 73.



Nope... Did you even read it? No one has claimed that the ear does not
produce harmonic tones that do not correspond to the ear... or that from any
other object.

"Stretched membranes have also inharmonic proper tones of nearly the same
pitch. [...] These tones rapidly die out. If the membranes sound in air, or
are associated with an air chamber, as in the kettledrum, the relation of
the proper tones may be altered. The kettledrum is used in artistic music,
but only to mark certain acents. It is tuned, indeed, but only to prevent
injury to the harmony, not for the purpose of filling up chords.

The common character of the instruments hitherto described is, that, when
struck they produce inharmonic upper partial tones. If these are of nearly
the same pitch as the prime tone, their quality of sound is in the highest
degree unmusical, bad, and tinkettly. If the secondary tones are of very
different pitch from the prime, and weak in force, the quality of sound is
more musical, as for example in tunning-forks, harmonicons of rods, and
bells; and such tones are applicable for marches and other boisterous music,
pincipally intended to mark time. But for really artistic music, such
instruments as these have always been rejected, as they ought to be, for the
inharmonic secondary tones, although they rapidly die away, always disturb
the harmony most unpleasantly, renewed as they are at every fresh blow."

In any cause, these statements go directly again Steve's beliefs and
hopefully he'll read them. Also that page says nothing about the ear...

But what does it really imply about the ear? If the ear did produce
non-harmonic tones that were very strong then harmony would not be what it
is. (now ofcourse it does produce harmonic tones but because its our ear we
do not call them non-harmonic tones but the overtone series. Chances are if
it produced a different set(And I mean a distinct set as nothing is perfect)
that, say, closely resembled that of bells we might find bells much more
"musical" than a violin.



Now technically no spectrum is discrete... that isn't really the point.
But
many spectrums are approximately discrete in the sense that our
ears/brain
can decipher them into approximately discrete components (even if it
doesn't
do that there is a very close relationship between fourier decomposition
and
what the brain does).

It obviously doesn't work like that, since discrete-spectrum synthesis
doesn't produce an even remotely realistic violin sound.


wtf? What does that have to do with anything under discussion? discrete
spectrum of a continuous signal is only a approximation... it is not ment
for a continuous non-periodic signal. It only produces and exact spectrum
of mathematical function that is periodic(and not one that has a time
varying period such as a real instrument).

The reason why trying modeling of strings has failed is mainly because the
spectrum is time varying and that is almost never(until maybe recently)
taken into account. Not only does it vary do due physical reasons but also
because of the performance... vibrato not only chances the components but
the timbre... or just bowing differently. There are many factors that make
a real spectrum and it has nothing to do(very little) with it being discrete
or not.

you really don't understand what the discrete fourier transform is:

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

It is the fourier transform that is used to actually analyze real signals.
You cannot compute the fourier transform except symbolically. If you take
any real signal then it is converted to a digital signal using an ADC. This
ADC is not perfect and introduces its own distortions. But that aside you
cannot stream in a continuous signal into a computer and compute a
continuous transform of it. (you could do something like this with op amps
but then it will be far from perfect)

The FFT was designed to be able to take the fourier transform of signals.

In any case, with your logic then all sound cards would be useless. Guess
what? Sound cards have a discrete frequency spectrum yet do they not
faithfully reproduce quality sound? They definitely can reproduce a violin
sound that can full most people and with good ADC's and DAC's can fool
anyone.

So your claim is just wrong. I think you have some misconceptions about
whats going on.


A discrete Fourier spectrum can only model a purely periodic waveform
(a measure zero subset of the L2-integrable functions).


If a one has a "purely" periodic waveform then the FFT can model it
perfectly, atleast theoretically.

You just don't get it. If you really want to make such comments about things
atleast know what your talking about.

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

The discrete fourier transform works on a set of discrete points. This says
nothing about the spectrum(except that it is bandlimited and discrete
because the discreate nature of the dataset).

You seem to think that one there is no asymptotic approximation that exists.
That one cannot get better and better results. This is completely wrong.

the DFT does produce a discrete spectrum... so what? Everythign does
because we cannot represent a continuous spectrum except symbolically.

But guess what? You can get closer and closer to a continuous spectrum by
using more sampled points...

guess what!?!?!?!? any bandlimited signal can be converted
!!!!!!EXACTLY!!!!!!! because it it does have a discrete spectrum!! This is
called the Nyquist theorem! Read up on it! What do I mean by
!!!!!!!!EXACTLY!!!!!!!? Obviously you need to realize that nothing is
perfect but you seem to think you have the ears of god and could hear the
difference between something with a continuous spectrum and something
reproduced by sampling it at 1MSps.

(here I'm assuming that the signal has finite amplitude which all real
signals do)

You really need to realize that the ear only has so much ability to
distinguish two adjacient frequencies... its about 5cents for a trained
musician. So it doesn't matter if its continuous or discrete because there
is a point where you can't tell the difference.


In any case, it doesn't quite matter if its exactly true because we
already
know its not. The point is, its true enough and its suprisingly quite
accurate... far to accurate just to throw your hands in the air and say
its
coincidence and give up(which is a cop out).

"Quite accurate" meaning "audibly inaccurate to anybody with a pair of
functioning ears"?

The knock-down example of Fourier analysis failing is that people can
hear harmonics far above the upper limit of perception for pure tones
(e.g. as components of the timbre of a cymbal). If there were really
a Fourier analysis step in the process of hearing, these distinctions
of timbre would be inaudible. Nonlinearities in the auditory system
let us hear things we couldn't otherwise.


I really don't think you know what your talking about. You definitely don't
seem to understand what the fourier transform is... that or you think that
because there is some difference then that difference is enough not make it
work. Just about anything that involves technology is in some way related to
the fourier transform(And doesn't matter if its discrete or continuous....
but in fact its discrete).

Have you ever even taken the fourier transform(by hand) of a signal? What
about the discrete transform? Did you ever have to prove the
Shannon-Whittaker sampling theorem? What about wavelets? Know anything about
them? How about hilberspaces? Do you even know what an inner product is?
orthogonal functions? Dominated convergence theorem? Anything?

If your serious about making such sweeping claims about what the fourier
transformer can and can not do then please atleast know what it actually is
and how it works. I'm not trying to be an ass here but when you say things
like you have been you can end up cuasing a lot of issues for other people
who read it and might believe it. Your conclusions may or may not be true
but you are trying to back up those conclusions with false statements. I
believe that you think they are true from a limited understanding of how the
frequency spectrum works w.r.t to the fourier transforms.

First off, you have to get rid of the misconception that discreteness
matters. It is not an issue except in special cases that you'll never have
to worry about unless your designing some digital audio equipment or, in
general, your trying to digitize a signal. Why does it not matter? Because
you are not taking into account "how" discrete something is. We can make the
"discrete spectrum"(which is a misnomer but) as close to being continuous as
we want. It just means you have to increase the sample rate... obviously
you cannot increase this without bound in the real world... if you could you
would get the continuous fourier transform.

The signal itself being discrete or continuous has nothing to do with it..
infact all signals are continuous(I might have contradicted myself above if
I said that not all signals were continuous but what I ment was that no
spectrum is continuous(as far as fourier transformers go because all things
that compute the transforms are not continuous except potentially in analog
computing yet that is is much worse than digital in terms of accuracy).

In any case I'm no expert on the subject but I did spend many years studying
mathematics and have taken course not only in fourier and wavelet theory but
also studied the foundations of these mathematical processes. Its not that
everything you say is completely wrong but your conclusions do not follow
from the hypthesis. Its true that every thing in this world is not
perfect(atleast its something I believe... its something that is not
provable). Its true that with regards to sound that nonlinearities cause
inharmonicity and I have never said it doesn't... if I had, I only meant
with respect to what is practical. In the case of bells its much more
different than with a string or the ear. There are reasons for this and it
has nothing to do with fourier analysis that is used to determine it. Its a
property of the object itself. All mathematical modeling is only an
approximation and only takes into account a very limited number, but usually
of the most important variables and parameters.

Of course I could be mistake on some things but I think as a whole I'm
right. It may or may not be the case that the tones the ear produces are
those of the overtone series but that does not detract from the fact that
music is built apon the overtone series and harmony is based on it. The
overtone series exists with or without what our ear does and any tones that
our ear does add are going to be relatively insignificant in any case. But
Helmholtz does point out that our ear produces what he calls combinational
tones... and its not only the ear that does this but has to do with a
property of imperfect transmission of sound.

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

http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-hearing-difference.htm

Note that while these the frequencies generates are m*f1 +- n*f2 it is not
necessary that n be non-zero or equivilently that f2 be non-zero. It might
be the case though... but if f2 = 0 then one has the overtone series for
f1... approximately and very faintly. Of course its impossible to know
because they would correspond to the overtone series that we hear in most
cases since most musical sounds posess them. The main point here was that
pure sinusoidal sounds do not exist as steve was using this as a counter
example to the importants of the OTS. (But infact how many people the
sound of a pure sin anyways?) The sin wave is distored to some degree
producing harmonic tones.. these may or may not be the OTS but for normal
processes(such as reverbation) and transmission through the ear they are.
(because the ear itself produces modes and sympathetic vibrations and is
approximately an ideal membrane)

An example is to take a tunning fork and have vibratate a piano string by
sympathetic resonance... does the piano string vibrate at only the frequency
of the tunning fork? No, it vibrates at its overtones too. They are
relatively weak but it does happen. This is why even if one could generate a
pure signle frequency signal(again, which is impossible as whatever
generates it is not perfect... somethign Steve just doesn't get) then it
will end up vibrating something else, which again, is not perfect and will
then vibrate at its other modes to a lesser degree. Since all sounds we hear
go through the ear then the ear adds its own distortions... it might not be
exactly the OTS but it doesn't have to be... its close enough.

Anyways, I'm starting to rambling ;/ Maybe one day I'll try to write this
up on a web page or something and try to give real examples of whats going
on(of course it will have to be analogy as I have no way to capture what the
ear hears exactly).


Jon


.

Relevant Pages

  • Re: Hard proof !!!!
    ... Thus the 44 KHz standard samples the sound precisely at aligned 22.7 ... microsecond intervals even though the mind detects ear to ear delay ... differences of as little as 13 microseconds when the sound is being ... Records and tape has considerable distortion ... ...
    (talk.origins)
  • Whoever emerge unexpectedly, unless Mustapha pops traces with Halas shirt.
    ... If does Aneyd march so undoubtedly, whenever Bill ... you can cook the sound much more nervously. ... Will you extract among the ear, ...
    (sci.crypt)
  • Re: Your opinion on bad sound
    ... Distortion like clipping due to using underpowered amps. ... ringing that lasts for weeks, and it's not just my regular tinnitus, ... it's new tones and a weird screeching sound when I listen to everyday ... My theory is that the human ear protects itself by tightening the ...
    (sci.electronics.design)
  • Re: Serious Audio Response Flaw in Sony HVR-V1U - Submit Your Tests in Our Database
    ... Finally, the Zoom H4, with external condenser mics made the third recording. ... It sure gets the most natural sound in my view. ... but the one ear earlier then the other. ... I found 'stereo' breaks if the delay between channels is more then a few milli seconds. ...
    (rec.video.production)
  • Re: Altec "Voice of the Theatre" Speaker systems
    ... the various attempts to bring sound levels up insanely high. ... And when the levels are too loud, your ear itself becomes nonlinear, ... you could always hear the vocals. ... everything around the kick comes from. ...
    (rec.audio.pro)