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Re: A new paradigm?(On pitch and periodicity (was "correction to post"))
I commend you on your bravery.
At 12:53 PM -0400 9/6/11, Ranjit Randhawa wrote:
I felt that it would not only be negligent on my part but also
cowardice, to end this thread without at least offering a possible
approach to a new "paradigm", even if I am not capable of describing
it in excruciating detail at the moment. So, with the greatest
trepidation, here goes.
If one were to consider a pure sinusoid in the phase domain (one
where the axis are x(t) and dx(t)/dt), the locus would be a circle.
The area of this circle would give us the magnitude, though how to
determine this requires a different approach as the integration over
2pi would be zero.
If we consider the product x(t)*dx(t)/dt as the rate of change of
energy it would have a sign associated with it, then it is possible
to determine this area, though the resulting algorithm would be too
simple and fall apart for more complex signals since we don't know
the period. To get a more general approach, it would be better to
consider the circle in sectors of harmonically increasing sizes,
thereby converting the sinusoid signal to a harmonic series, the
area of each sector becomes the magnitude of a related tonal
harmonic with the smaller sectors associated with magnitudes of
We see then that we start with a single sinusoid being considered
not as a single valued entity but as a harmonic series and this
therefore immediately answers two questions, the first being the
reported psycho-acoustical behavior whereby some people have
indicated an ability to recognize harmonics of a pure tone and shown
by others to be possible by using beats. The second being that the
extent of the traveling wave can have an explanation in that the
stiffness of the BM would limit activity for higher frequencies as
these frequencies would have smaller areas for lower strength
signals and the TW would grow as this strength increases.
More importantly, since we are directly using energy to determine
magnitudes, the missing fundamental would show a magnitude and this
magnitude would vary depending upon the relative phases of the
components. This approach also provides for a much reduced
computational method to determine the period, an alternative form of
auto-correlation. These two assertions are based on the method
chosen for determining the harmonic series and the one chosen by me
was picked from the field of psychology and called "evaluative
bivalence", whereby one makes use of sign associated with the rate
of change of energy in the summation process.
The modified auto-correlation does work rather well for
quasi-periodic signals and would welcome any suggestions for
practical use, since I don't believe it is used by the auditory
system. I believe it would fail in the "party room" environment.
Other options are possible based on the summation process.
There are many consequences of this approach as it now becomes
possible to provide a more exact method to explain source location
capabilities, pitch explanations, and I would like to say cochlear
functions but have to admit my knowledge at that level is focused
only on what has been reported on the behavior of the Traveling
Wave. The rest of it is a mystery to me.
I would like to apologize if this blurb causes some kind of angst
among some in this LIST. It was not the intent. I simply wanted to
show that sticking with existing mathematics has not made much
progress in being to explain our original discussion and that was
"The Case of the MIssing Fundamental". Thanks for your understanding
and kindness and sorry for this delay,
On 8/4/2011 1:42 PM, Richard F. Lyon wrote:
I'll be the first to agree that linear systems theory is sometimes
stretched beyond where it makes sense, and that you need to use
nonlinear descriptions to describe pitch perception and most other
aspects of hearing, and more so when you get up to cognitive levels.
I'm sorry to hear that you "gave up on linear systems", because I
don't think it's possible to do much sensible with nonlinear
systems when you don't have linear systems as a solid base to build
on. Certainly at the level of HRFTs, cochlear function, and pitch
perception models, a solid understanding of linear systems theory
is in indispensible prerequisite. Then, the nonlinear
modifications needed to make better models will seem less
At 10:33 AM -0400 8/4/11, Ranjit Randhawa wrote:
While linear system theories seem to work reasonably well with
mechanical systems, I believe they fail when applied to Biological
systems. Consider that even Helmoholtz had to appeal to non-linear
processes (never really described) in the auditory system to
account for the "missing fundamental" and "combination tones".
Both of these psycho-acoustical phenomenon have been well
established and explanations for pitch perception are either
spectral based or time based with some throwing in learning and
cognition to avoid having to make the harder decision that maybe
this field needs a new paradigm. This new paradigm should be able
to provide a better model that explains frequency (sound!)
analysis in a fashion such that the nothing is missing and
parameter values can be calculated to explain pitch salience, a
subject that seems to be never discussed in pitch perception
Furthermore, such a new approach should also be able to explain
why the cochlear is the shape it is, which as far as I can see has
never been touched upon by existing signal processing methods.
Finally, are these missing components "illusions" that are filled
in so to speak by our higher level cognitive capabilities? It is
remarkable that this so called filling in process is as robust as
it is, to be more or less common to everyone, and therefore one
wonders if all the other illusions are really not illusions but
may have a perfectly good basis for their existence. If they were
"illusions" one would expect a fair amount of variation in the
psycho-acoustic experimental results I would think.
I myself gave up on linear systems early in my study of this field
and have felt that other systems, e.g. switching, may offer a
better future explanatory capability, especially when it comes to
showing some commonality of signal processing between the visual
and the auditory system. To this end, I am quite happy to accept
that I do not consider myself an expert in linear system theory.
On 8/2/2011 1:49 PM, Richard F. Lyon wrote:
At 5:55 PM +0300 8/2/11, ita katz wrote:
The periodicity is determined by the least-common-multiple of
the periodicities of the present harmonics, so if (for example)
a sound is composed of sines of frequencies 200Hz, 300Hz, and
400Hz, the periods are 5msec, 3 1/3msec, and 2.5msec, so the
least-common-multiple is 10msec (2 periods of 5msec, 3 periods
of 3.33msec, and 4 periods of 2.5msec), which is of course the
periodicity of the sum of the sines, or in other words 100Hz.
(actually it is the same as the greatest-common-divisor of the
Ita, that explanation is sort of OK, but as written implies that
the auditory system has the ability to do number-theory
operations on periods (or frequencies), and depends on there
being harmonics present and separately measureable.
It would be much more robust to say that "The pitch is determined
based on an approximately common periodicity of outputs of the
cochlea," which I believe is consistent with your intent.
Why is this better? First, it doesn't say the periodicity is
determined; what is determined is the pitch (even that is a bit
of stretch, but let's go with it). Second, it doesn't depend on
whether the signal is periodic, that is, whether harmonics exist.
Third, it doesn't depend on being able to isolate and separately
characterize components, harmonic or otherwise. Fourth, it
doesn't need "multiples" (or divisors), but relies on the
property of periodicity that a signal with a given period is also
periodic at multiples of that period, so it only needs to look
for "common" periodicities--which doesn't require any arithmetic,
just simple neural circuits. Fifth, it admits approximation, so
that things like "the strike note of a chime" and noise-based
pitch can be accommodated. Sixth, it recognizes that the cochlea
has a role in pitch perception. It's still not complete or
perfect, but I think presents a better picture of how it actually
works, in a form that can be realistically modeled.
Is this "tortured use of existing signal processing techniques"
as Randy puts it? I don't think so. Is it "a unique way to do
frequency analysis and to meet the dictum in biology that 'form
follows function'"? Sure, why not? But why call it "frequency
analysis"? How about "a unique way to do sound analysis" (if by
"unique" we mean common to many animals)?
I do have some sympathy for Randy's concern that we are far from
a complete understanding, and that hearing aids are not as good
as they would be if we understood better, but yes, he sounds way
too harsh in overblowing it so. I'm wondering what's behind
that, and whether it's just confusion about all the confusing
literature on pitch perception, which I agree is a complicated
mess -- or is the problem, indicated by Randy's previous posts,
just that he doesn't understand basic linear systems and signal
processing, and that's why it all seems "tortured"?