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on url What is Octave Equivalence?

in an answer on the related ( octave equivalence ) question is stated:

Mathematically, this signifies that the mammalian cochlea differentiates and squares the incoming sound pressure signal.

In terms of physics, it means that a sound energy signal is offered to the organ of Corti. Functioning as a Fourier analyzer, the organ of Corti subsequently converts these incoming signals into the sound energy frequency spectrum that is transferred to the auditory cortex in a frequency selective way.

Salient experimental results so far • For residual tone complexes – harmonic series where the first harmonic or fundamental is missing – the differentiating and squaring process in the cochlea reconstructs perfectly the corresponding but missing fundamental. • Contrary to the conclusion that an early neural mechanism is responsible for the mystery of the inferential pitch, strong evidence exists that the cause for this reconstruction of the virtual or fundamental pitch is hydrodynamic in origin.

and on that answer / statement:

"For residual tone complexes – harmonic series where the first harmonic or fundamental is missing – the differentiating and squaring process in the cochlea reconstructs perfectly the corresponding but missing fundamental."

a user reported a dichotic stimulation experiment and staded:

It has been known for over 70 years that this phenomena is not occuring exclusively within the cochlea. Listening tests have been performed in which two sine tones are played to the individual ears of a listener, and yet the listener is still able to identify an underlying fundamental frequency. Hence, there is definitely some "mixing" distortion taking place in the brain.

Question:

Can the differentiating and squaring process in the cochlea explain the reported dichotic stimulation experiment ?


I fail to see any mathematical content in this question. To turn it into a mathematical question, you'd have to say a lot more about that process and the assumptions you're making. – joriki


Ok, here I say some more. The mathematical differentiating and squaring process in the cochlea is explained in the booklet "Applying physics makes auditory sense".

http://igitur-archive.library.uu.nl/med/2011-0204-200555/UUindex.html

And it is about the comment on: "For residual tone complexes – harmonic series where the first harmonic or fundamental is missing – the differentiating and squaring process in the cochlea reconstructs perfectly the corresponding but missing fundamental."

where a user commented this and stated:

It has been known for over 70 years that this phenomena is not occuring exclusively within the cochlea. Listening tests have been performed in which two sine tones are played to the individual ears of a listener, and yet the listener is still able to identify an underlying fundamental frequency. Hence, there is definitely some "mixing" distortion taking place in the brain.

  • And that is a useful comment to consider.

I am affiliated with this book. I am the co-author of the Appendices.

Year Article Author(s) Source 2010 1 Applying physics makes auditory sense : a new paradigm in hearing Abstract | Full-text Heerens, W.C., Ru, J.A. de Medicine (2010), pp: 1-74

I have had contact with Willem Heerens.

Based on the new cochlear model that is described by Heerens in his booklet: “Applying Physics Makes Auditory Sense”, he explained to me that the above reported dichotic stimulation experiment – two sine tones played to the individual ears of a listener – is just one of the following series of experiments which can be extremely well explained by what he names: “hydro dynamical crosstalk between both ears”.

He explained to me that it works as follows:

Under normal conditions every stimulus of the one ear – i.e. perilymph push-pull inside the cochlea – will be transferred for a small part via the two cochlear aqueducts and the cerebrospinal cavity of the skull to the other ear [most of the push-pull occurs between the oval window and the round window].

Due to the place where the entrance of the cochlear aqueduct in the scala tympani is situated this dichotic transfer is mainly in opposite phase [based on the time delay due to the travel time of the signal from ear to ear the signal’s phase delay increases linearly with increasing frequency].

And after arriving from the one ear in the other ear the stimulus will become a contribution to the perilymph push-pull and hence will be part of differentiating and squaring process.

Automatically the sound energy stimulus of the basilar membrane will be generated by the non-stationary Bernoulli Effect and the sum and difference frequencies of the two primary stimuli will be evoked as well, next to the two doubled frequencies.

And now we can look at a number of experiments:

  1. The Huggins pitch.

    If we give the one ear a stimulus of white or pink noise and the other ear the same stimulus except for a 180° phase shift in a small frequency domain around approx. 600 Hz, the listener will hear a faint tone of 600 Hz.

    Explanation: All the frequencies – both lower and higher than the small 600 Hz domain – that arrive from the one ear via the cerebrospinal cavity into the other are almost in opposite phase. Hence they will reduce the summated stimulus in each ear.

    However the mutual frequency contributions close to 600 Hz are pretty well in phase. And they will increase the summated stimulus around 600 Hz in each ear. The result is that above the noise signal in each ear a faint but hearable 600 Hz tone is evoked.

  2. The Binaural Masking Level Difference.

    See for instance: Brian C.J. Moore: “An Introduction to the Psychology of Hearing”.

    The following happens:

    A pure tone – low frequency and not higher than 1500 Hz – in combination with white noise is fed identical to both ears.

    The loudness of the tone is reduced to the level of “just masked by the noise” stimulus.

    a. Changing the phase with 180° of one of the two tones while the noise stimuli remain unchanged makes the tone hearable again in both ears. The tone rises above the noise level in both ears.

    b. Changing the phase of the noise stimulus in one ear with 180° while the two tone contributions remain in identical phase makes that the tone will be heard again in both ears. Because the noise level is lowering.

    If we stimulate only one ear with the tone masked by the noise, the following happens:

    c. The tone won’t be heard in both ears of course.

    d. However if we stimulate the other ear with the same noise stimulus as the firstly stimulated ear the tone will be heard again in that ear. The noise masking level is lowered and the tone is heard again – but in one ear.

    Also here the explanation is found in the either in of off phase effects in the transferred stimuli via the cerebrospinal cavity.

  3. Binaural Beat Phenomena.

    The same hydro-dynamic behavior exists for binaural beat stimuli. The only difference with the experiments – actually the so called Diana Deutsch illusions – described is that the two frequencies applied have a low (approx. 10 Hz) frequency difference.

    And again: No mixing in the brain, but pure ‘hydro dynamical crosstalk’.

    It is simply a matter of observing the cochlear functioning in a completely different but realistic way.

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I fail to see any mathematical content in this question. To turn it into a mathematical question, you'd have to say a lot more about that process and the assumptions you're making. –  joriki Jan 8 '13 at 14:20
    
I edited the question to say a little more. I highlighted differentiating and squaring in the text and accordingly sum and difference frequencies and doubled frequencies. The now mentioned booklet provides the mathematics. And well also, I disclosed my affiliation. –  Yves Mangelinckx Jan 8 '13 at 17:00

1 Answer 1

up vote 1 down vote accepted

Given a Fourier series for the pressure $P$ that is missing the fundamental tone ($e^{\pm i t}$ in this case), $$P=\cdots+ d^*e^{-4 i t} + c^*e^{-3 i t} + b^* e^{-2 i t} + b e^{2 i t} + c e^{3 i t} + d e^{4 i t} +\cdots$$ then we get $$\begin{align} [dP/dt]^2&=\cdots %+e^{-8 i t} \left(-16 \left(d^*\right)^2+\cdots\right) %+e^{-7 i t} (-24c^* d^*+\cdots) %+e^{-6 i t} \left(-16 b^* d^*-9 \left(c^*\right)^2+\cdots\right) %+e^{-5 i t} (-12 b^* c^*+\cdots) %+e^{-4 i t} (-4 \left(b^*\right)^2+\cdots) +e^{-2 i t} (16 b d^*+\cdots) +\color{red}{e^{-i t} \left(12 b c^*+24 c d^*+\cdots\right)}\\ & + \;(8 b b^* + 32 d d^* + 18 c c^* +\cdots) +\color{red}{e^{i t} \left(12 c b^*+24 d c^*+\cdots\right)} +e^{2 i t} (16 d b^*+\cdots) %+e^{4 i t} (-4 b^2+\cdots) %+e^{5 i t} (-12 b c+\cdots) %+e^{6 i t} \left(-16 b d-9c^2+\cdots\right) %+e^{7 i t} (-24 c d+\cdots) %+e^{8 i t}(-16 d^2+\cdots) \end{align}$$ We see we get terms in red corresponding to the missing fundamental, provided the coefficients converge to a nonzero number. For normal sounds that may be true. I'm afraid I don't know in general...

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