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Re: A new paradigm?(On pitch and periodicity (was "correction to post"))
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,
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 higher frequencies.
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
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
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
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 "tortured".
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
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"?