# Re: A new paradigm?(On pitch and periodicity (was "correction to post"))

Hi Matt,

just to be clear: I only used my pond analogy to indicate that the traveling wave observed on the surface is caused by the traveling wave in the fluid. In both cases there is a fluid domain with a flexible boundary with a restoring force. I never meant to imply that surface waves on a pond can be used to describe the details of what's going on in the cochlea. The physical processes and forces at work in the two cases are completely different. Sorry if this was confusing.

Peter

2011/9/22 Matt Flax
Indeed,

No confusion here by the way !

My deep water tsunami analogy is nothing like your pond analogy. Your
pond analogy only really makes sense at the base of the cochlea.

Travelling waves are generally regarded as having differential pressure
about the BM - pressure waves are generally regarded as having equal
pressure about the BM. The 'reflection' is a minor response (mode) to
the active apex and a major response mode to the active base.

Our mixed mode Cochlear Amplifier model clearly states this.

Matt

On Thu, 2011-09-22 at 09:28 +0200, Peter van Hengel wrote:
> Hi Matt and list members,
>
> thanks for the support. One small comment to your statement that there
> are only forward traveling waves. I think much of the confusion stems
> from the fact that the observed motion of the cochlear partition is
> often referred to as the traveling wave. In actual fact this is only a
> 'reflection' of the actual wave which is traveling in the fluid. The
> fluid supports waves traveling in both directions, but the unique
> properties of the cochlea cause only the appearance of a forward
> traveling on the cochlear partition.
> If one wants to observe the reverse traveling waves in the cochlea it
> is necessary to measure fluid velocity, which I believe is not yet
> possible. Model calculations clearly show the reverse traveling wave
> and produce results in accordance with data on OAEs (see e.g. the work
> of Mauermann et al or Epp et al). But I'll (re)check the work of Ren
> to make sure I'm not relying only on what I believe to be true ;-).
> Thanks also to Charles for the support. And many thanks to Dick for
> starting the effort!
>
> All the best,
> Peter
>
> 2011/9/21 Matt Flax <flatmax@xxxxxxxx>
>         I am glad you brought this up again Peter,
>
>         I have no problem with the passive travelling wave, however
>         with respect
>         to actively induced movements, the latest experimental data
>         shows that
>         there are ONLY forward travelling waves (check Ren's
>         experiments for
>         example).
>
>         OAEs are generated locally (this is common belief)
>         consequently one
>         assumes that the basilar membrane is not on the surface, but
>         suspended
>         inside the pond and the oval/round windows are on the surface
>         of the
>         pond.
>
>         What's more, if the OHCs in the apex are regarded as deep
>         ocean
>         movements, then the waves resemble tsunamis ! Consequently
>         only
>         movements near to the shore/surface (the windows) are easily
>         observable
>         - the not easily observable being the small movements of the
>         OHCs in the
>         apex themselves and the pressure waves (large in potential but
>         tiny in
>         flux or velocity) in the incompressible fluid.
>
>         Matt
>
>
>         On Wed, 2011-09-21 at 12:21 +0200, Peter van Hengel wrote:
>         > Dear dr Heerens and list-members,
>         >
>         > I hesitate to get involved in this discussion as I have
>         tried to
>         > explain the hydrodynamics behind (transmission line) cochlea
>         models
>         > before in another thread on this list and don't like
>         repeating myself.
>         > But I feel I have to lend my support the comments made by
>         Dick Lyon.
>         > As I have stated before fluid physics states that a fluid
>         domain (such
>         > as the cochlea or a pond) with a flexible boundary subject
>         to a
>         > restoring force (such as the aochlear partition or the pond
>         surface)
>         > MUST exhibit 'ripples' on the surface. In the cochlea these
>         are
>         > refered to as traveling waves. The wave energy is not
>         traveling in the
>         > boundary itself but in the fluid. Any attempts to prove that
>         such
>         > waves do not exist, or are based on 'bad physics', are
>         unfortunately
>         > based on a lack of understanding of the fluid mechanics.
>         > Whether the traveling wave is the only mechanism responsible
>         for
>         > transporting sound energy to the hair cells is still a valid
>         question,
>         > but untill an alternative model produces similar or better
>         results on
>         > modeling physiological, pshychophysical and OAE date, I'll
>         stick with
>         > the transmission line. Things like pitch perception and the
>         missing
>         > fundamental can perhaps not be explained purely by looking
>         at the
>         > average excitation caused by the traveling wave, but I don't
>         think
>         > anyone ever claimed they could. In my opinion it is good to
>         develop
>         > new theories, but we should attempt to integrate them with
>         existing
>         > ones instead of throwing away something that has proven to
>         work.
>         >
>         > Kind regards,
>         > Peter van Hengel
>         >
>         >
>
>         > 2011/9/19 Willem Christiaan Heerens <heerens1@xxxxxxxxx>
>
>
>         >         Dear Dick Lyon,
>         >
>         >
>         course I
>         >         With pleasure.
>         >
>         >         You wrote:
>         >         **Sometimes it's hard to get a reaction when you are
>         trying to
>         >         replace a
>         >         paradigm, as the silence here illustrates.  I didn't
>         really
>         >         get into the
>         >         new ideas of your book much, but I have some
>         >         introductory
>         >
>         >         material about why you reject the current
>         >
>         >         Your reaction in the first sentence is pretty well
>         familiar to
>         >         me. It is
>         >         entirely in accordance with the procedure described
>         by Thomas
>         >         Kuhn in his
>         >         world famous 1962  essay:
>         >
>         >         “The Structure of Scientific Revolutions”
>         >
>         >         Besides that: a former colleague of mine, a highly
>         skilled
>         >         senior professor
>         >         in applied physics, who reviewed our booklet during
>         a
>         >         contribution
>         >         procedure for a scientific journal, quite recently
>         gave us the
>         >         verdict that
>         >         he fully agreed with our arguments and statements
>         and he urged
>         >         the editor
>         >         to make a full scientific discussion possible for
>         our views.
>         >         He also warned
>         >         me that to be in right is not the same as to be put
>         in right.
>         >         I myself
>         >         don’t see all this as a problematic issue. It’s part
>         of the
>         >         way messengers
>         >         or designers of new paradigms are encountered by the
>         mayor
>         >         supporters of
>         >         the competing one. Of course the scientific
>         reputation
>         >         rankings of so many
>         >         scientists are involved and in danger in case a
>         >         is
>         >         happening.
>         >
>         >         The only issue that counts for me is that scientific
>         arguments
>         >         from both
>         >         sides brought in discussion, verified and weighted
>         in a
>         >         careful way must
>         >         turn the balance. Ignoring irrefutable arguments
>         because they
>         >         for the ranking of a scientist has always been
>         contra
>         >         productive for the
>         >         progress in a field of science. History shows many
>         of such
>         >         examples. One of
>         >         the most salient among them certainly is the
>         Copernican
>         >         revolution.
>         >
>         >         The result of the second line of your comment I
>         really regret,
>         >         because in
>         >         the rest of your writings I clearly can see that you
>         have
>         >         apparently
>         >         missed, misread or misinterpreted a number of issues
>         on
>         >         cardinal points.
>         >
>         >         Let me discuss your next comment:
>         >
>         >         **You discuss and reject two wave concepts: first,
>         the
>         >         pressure sound wave
>         >         that travels so fast that wavelengths will always be
>         long
>         >         compared to the
>         >         size of the cochlea, and second, "capillary" or
>         "interfacial"
>         >         waves,
>         >         presumably meaning those water surface waves where
>         gravity
>         >         provides the
>         >         restoring force.  Of course, neither of these can be
>         the
>         >         explanation for
>         >
>         >         how the cochlea works.**
>         >
>         >         I don’t reject the pressure sound wave concept, at
>         least not
>         >         in general. It
>         >         is of course the vehicle of mechanical vibration
>         energy and
>         >         therefore also
>         >         acoustical vibration energy. How could an academic
>         physics
>         >         scientist reject
>         >         that?
>         >         What I have argued is that for all the frequencies
>         that can be
>         >         sensed in
>         >         the cochlea even up to 20 kHz counts that the sound
>         velocity
>         >         in perilymph –
>         >         being 1500 m/s – in relation with these frequencies
>         result in
>         >         a wave length
>         >         always larger than 75 mm.
>         >         So therefore this mechanism cannot contribute to a
>         >         discriminating mechanism
>         >         for frequency selectivity based on traveling waves.
>         >
>         >         And regarding the  "capillary" or "interfacial"
>         waves I
>         >         reject: yes indeed
>         >         in quite a number of textbooks I see the comparison
>         of the
>         >         propagation of
>         >         surface waves in a pond with the slow waves inside
>         the
>         >         cochlea. It simply
>         >         is an erroneous analogon. None of the parameters
>         necessary for
>         >         the
>         >         existence of capillary waves can be found inside the
>         cochlea.
>         >         So neither
>         >         they can play a role in evoking traveling waves that
>         have
>         >         short wavelengths.
>         >
>         >         You wrote:
>         >
>         >         **You also attribute to Lighthill some strange wrong
>         ideas
>         >         transmission lines only being able to transmit
>         energy near
>         >         their resonance.
>         >
>         >         **
>         >
>         >         Can you be more specific?  The only lines I describe
>         are the
>         >         lines in Fig.
>         >         1. That figure is a reproduction of the figure in
>         Lighthill’s
>         >         paper:
>         >
>         >         Lighthill MJ. (1981) Energy flow in the cochlea. J
>         Fluid Mech
>         >         106: 149-213.
>         >
>         >         I haven’t attributed strange wrong ideas to
>         Lighthill. I have
>         >         studied
>         >         carefully all the 64 pages  of his paper.
>         >
>         >         He starts with a very informative series of premises
>         and I
>         >         cite this part:
>         >
>         >         *** With moderate acoustic stimuli, measurements of
>         >         basilar-membrane
>         >         vibration (especially, those using a Mössbauer
>         source attached
>         >         to the
>         >         membrane) demonstrate:
>         >         (i) a high degree of asymmetry, in that the response
>         to a pure
>         >         tone falls
>         >         extremely sharply above the characteristic
>         frequency, although
>         >         much more
>         >         (ii) a substantial phase-lag in that response, and
>         one which
>         >         increases
>         >         monotonically tip to the characteristic frequency;
>         >         (iii) a response to a 'click' in the form of a
>         delayed
>         >         'ringing'
>         >         oscillation at the characteristic frequency, which
>         persists
>         >         for around 20
>         >         cycles.
>         >         This paper uses energy-flow considerations to
>         identify which
>         >         features in a
>         >         mathe¬matical model of cochlear mechanics are
>         necessary if it
>         >         is to
>         >         reproduce these experi¬mental findings.
>         >         The response (iii) demands a travelling-wave model
>         which
>         >         incorporates an
>         >         only lightly damped resonance. Admittedly, waveguide
>         systems
>         >         including
>         >         resonance are described in classical applied
>         physics. However,
>         >         a classical
>         >         waveguide resonance reflects a travelling wave, thus
>         >         converting it into a
>         >         standing wave devoid of the substantial phase-lag
>         (ii); and
>         >         produces a low-
>         >         frequency cut-off instead of the high –frequency
>         cut-off (i).
>         >         By contrast, another general type of travelling-wave
>         system
>         >         with resonance
>         >         has become known more recently; initially, in a
>         quite
>         >         different context
>         >         (physics of the atmosphere). This is described as
>         >         critical-layer resonance,
>         >         or else (because the reso¬nance  absorbs energy)
>         >         critical-layer absorption.
>         >         It yields a high-frequency cut-off; but, above all,
>         it is
>         >         characterized by
>         >         the properties of the energy flow velocity. This
>         falls to zero
>         >         very steeply
>         >         as the point of resonance is approached; so that
>         wave energy
>         >         flow is
>         >         retarded drastically, giving any light damping which
>         is
>         >         present an
>         >         unlimited time in which to dissipate that energy.
>         >         Existing mathematical models of cochlear mechanics,
>         whether
>         >         using one-, two-
>         >          or three-dimensional representations of cochlear
>         geometry,
>         >         are analysed
>         >         from this standpoint. All are found to have been
>         successful
>         >         (if only light
>         >         damping is incorporated, as (iii) requires) when and
>         only when
>         >         they
>         >         incorporate critical-layer absorption. This resolves
>         the
>         >         certain grossly unrealistic one-dimensional models
>         can give a
>         >         good
>         >         prediction of cochlear response; it is because they
>         >         incorporate the one
>         >         dimensional feature of critical-layer absorption.***
>         >
>         >         Apparently Lighthill has never considered the
>         possibility that
>         >         the observed
>         >         movements of the basilar membrane could be caused by
>         another
>         >         phenomenon
>         >         than a sound energy transporting traveling wave.
>         >
>         >
>         >         **Actually, he showed the opposite:  that a
>         sinusoidal wave
>         >         will propagate
>         >         until the point where the transmission line
>         resonance gets low
>         >         enough to
>         >         match the wave frequency, and at that point it will
>         slow down
>         >         to zero
>         >         velocity and die out.  This is not exactly how the
>         cochlea
>         >         works (the BM is
>         >         not very resonant), but not a bad concept from base
>         to near
>         >         the best
>         >         place.**
>         >
>         >
>         >         You say it clearly enough:   ‘It isn’t a bad concept
>         from base
>         >         to near the
>         >         best place.’
>         >         So not having an exact agreement between theory and
>         practice
>         >         makes the
>         >         underlying hypothesis directly vulnerable for
>         falsification.
>         >
>         >         Indeed the cochlea cannot react like that. And I
>         want to make
>         >         this clear by
>         >         the following series of experiments:
>         >
>         >         Entirely based on the premises of the new paradigm I
>         have
>         >         described, I now
>         >         have calculated a number of predictable sound
>         phenomena by
>         >         using the
>         >         following frequencies together with prescribed phase
>         relations
>         >         in a
>         >         standard summation procedure to compose a Fourier
>         series:
>         >
>         >         1:
>         >                    10000 + 10004 + 10008 + 10012 + 10016 +
>         10020 +
>         >         10024 Hz
>         >                     Where all the contributions are sine
>         functions.
>         >
>         >         Our paradigm predicts:  an undisputable beat of 4 Hz
>         in a high
>         >         beep tone.
>         >
>         >         2:
>         >                    10000 + 10004 + 10008 + 10012 + 10016 +
>         10020 +
>         >         10024 Hz
>         >                    Where the contributions are successively
>         >         alternating sine and
>         >         cosine functions.
>         >
>         >         Our paradigm now predicts:  an undisputable beat of
>         8 Hz in
>         >         the same high
>         >         beep tone.
>         >
>         >         3:
>         >                    10000 + 10004.0625 + 10008 + 10012.0625 +
>         10016 +
>         >         10020.0625 +
>         >         10024 Hz
>         >                    Where all the contributions are sine
>         functions.
>         >
>         >         Our paradigm now predicts:  a  beep, in which an
>         undisputable
>         >         beat exists
>         >         that changes every 8 seconds from clearly 4 Hz to 8
>         Hz and
>         >         then reverses
>         >         again to 4 Hz. So the beat pattern has a period of 8
>         seconds
>         >         caused by the
>         >         systematic mistuning of 1/16 = 0.0625 Hz.
>         >
>         >         Additional changes in the mistuning, like for
>         instance from
>         >         10004.0625 into
>         >         10003.9375 Hz, of either one, two or three of the
>         mistuned
>         >         frequencies are
>         >         predicted to give the same results in the beat
>         pattern as
>         >         experiment 3.
>         >
>         the software
>         >         program of
>         >         Yves Mangelinckx  with which these sound complexes
>         can be
>         >         properly
>         >         calculated in the form of wav files from the
>         following site:
>         >
>         >
>         http://www.a3ccm-apmas-eakoh.be/a3ccm-apmas-eakoh-index.htm
>         >
>         >         [ NOTE:    The standard setting in the 1/f mode in
>         this
>         >         software program
>         >         takes care that all the individually primary
>         calculated
>         >         frequencies
>         >         contribute equal energy to the resulting sound
>         pressure
>         >         signal. This
>         >         condition is very important for the influences on
>         pitch
>         >         calculations in
>         >         case higher values of the differences between
>         contributing
>         >         frequencies
>         >         exist. ]
>         >
>         >         This in order to give the interested reader the
>         opportunity to
>         >         falsify or –
>         >         in case our predictions are correct – to verify our
>         findings.
>         >
>         >         And of course I wouldn’t have given these examples
>         if I wasn’t
>         >         sure of my
>         >         statements.
>         >         I can already inform you that verification will be
>         the result.
>         >
>         >         If you carry out the same series of experiments with
>         a start
>         >         frequency of
>         >         1000 Hz instead of 10000 Hz, you will hear the same
>         series of
>         >         beat
>         >         phenomena, but now with the lower beep of the 1012
>         >         of the 10012
>         >         Hz beep.
>         >         Even if you go down with the start frequency to 200
>         Hz or 400
>         >         Hz you will
>         >         still hear the same beat phenomena, but now with the
>         low
>         >         humming tone of
>         >         200 Hz respectively with the one octave higher
>         humming tone of
>         >         400 Hz.
>         >
>         >         Hence it is a perception phenomenon that appears all
>         over the
>         >         entire
>         >         auditory frequency range.
>         >
>         >         And it must be remarked that according to the
>         current hearing
>         >         theory all
>         >         the used frequencies – especially in the higher
>         frequencies
>         >         like in the
>         >         10000 Hz experiments – according to auditory
>         experts, and also
>         >         supported by
>         >         Lighthill, will propagate by means of a traveling
>         wave to one
>         >         and the same
>         >         location on the basilar membrane.
>         >
>         >         If we then still follow the current hearing
>         >         to believe
>         >         that the medley of that seven totally unresolved
>         frequencies
>         >         will be
>         >         transferred via one and the same nerve fiber to a
>         location in
>         >         the auditory
>         >         cortex, where finally out of this ‘Gordian knot of
>         stimuli’ a
>         >         beep with the
>         >         described and also heard beat patterns will be
>         reconstructed.
>         >
>         >         Once these beat phenomena are verified as really
>         existing for
>         >         every
>         >         listener with a reasonable normal hearing, do you
>         agree with
>         >         me that for
>         >         the current paradigm this is a very serious anomaly?
>         >         In my opinion forcing an explanation within the
>         framework of
>         >         the current
>         >         paradigm will result in such a complexity that the
>         general
>         >         rule in science,
>         >         known as  ‘Ockham’s Razor’, to strive to an optimum
>         in
>         >         simplicity will be
>         >         strongly violated.
>         >
>         >
>         >         **You conclude that "the existence of two sound
>         energy
>         >         transport phenomena
>         >         with different transfer velocities within this tiny
>         cochlear
>         >         volume of
>         >         perilymph fluid as suggested by Lighthill is
>         impossible."  Yet
>         >         all
>         >         observations do see a slow wave, much slower than
>         the speed of
>         >         sound, and
>         >         basic mathematical physics of the same sort that has
>         been
>         >         working well for
>         >         over 100 years to describe waves in fluids predicts
>         exactly
>         >         that behavior.
>         >         Some may quibble that it has not been conclusively
>         proved that
>         >         the observed
>         >         slow wave carries energy; but no workable
>         alternative has been
>         >         put forward,
>         >         and no experiment convincingly contradicts this main
>         >         hypothesis of the
>         >         current paradigm, as far as I know.  I know some on
>         this list
>         >         will probably
>         >
>         >         say I'm wrong, now that I've opened the door.**
>         >
>         >         Do you agree with me that the perilymph inside the
>         cochlear
>         >         duct, existing
>         >         of scala vestibuli and scala tympani, is just moving
>         back and
>         >         forth over
>         >         distances not exceeding a few micrometer?
>         >
>         >         If you admit this fact, you should also agree with
>         me that all
>         >         the known
>         >         and involved physical quantities and parameters
>         indicate that
>         >         we are
>         >         confronted here with the problem to find the
>         hydrodynamic
>         >         solution for the
>         >         non-stationary small movements of an incompressible
>         >         non-viscous fluid in a
>         >         tiny narrow duct.
>         >         According to the rules of physics it is then
>         permitted without
>         >         any
>         >         additional constraints to use the non-stationary
>         Bernoulli
>         >         equation.
>         >
>         >         The exact and detailed solution of this equation I
>         can – if
>         >         you wish – send
>         >         you separately.
>         >
>         >         The result is exactly the mathematical _expression_ I
>         have used
>         >         in the
>         >         booklet:  the pressure decrease in the perilymph
>         duct in front
>         >         of the
>         >         basilar membrane is everywhere proportional to the
>         perilymph
>         >         velocity
>         >         squared.
>         >         What leads to the overall result that the pressure
>         stimulus on
>         >         the basilar
>         >         membrane is proportional to the sound energy
>         stimulus offered
>         >         to the ear.
>         >
>         >         You further wrote:
>         >
>         >                   **Yet all observations do see a slow wave,
>         much
>         >         slower than the
>         >         speed of sound.**
>         >
>         >         Indeed, an observation of a ‘slow wavy movement’ and
>         the only
>         >         place where
>         >         we can observe this is the basilar membrane.
>         >
>         >         It isn’t the occurrence of a wavy movement
>         phenomenon that we
>         >         have to
>         >         discuss. It is the origin of that ‘traveling wave’
>         that we
>         >         have to
>         >         discover. Is it a vibration energy transporting wave
>         or is it
>         >         a phase wave,
>         >         originated out of the manner in which the resonators
>         in the
>         >         basilar
>         >         membrane are grouped?
>         >
>         >         By the way, that is also – but not in an extended
>         way –
>         >         explained in our
>         >         booklet. In that chapter of the booklet I describe
>         why those
>         >         ‘waves’ always
>         >         run from base to apex. It is conform to the peculiar
>         mechanics
>         >         of the
>         >         basilar membrane system that this phase wave
>         behavior is
>         >         prescribed as it
>         >         is.
>         >         And that mathematical solution for this mechanics
>         problem of
>         >         resonators –
>         >         in case of the logarithmical frequency distribution,
>         low near
>         >         the apex to
>         >         high near the base – can be calculated, as I have
>         done,
>         >         analytically for a
>         >         pure sinusoidal tone, which exactly results in a
>         tonotopical
>         >         symmetrical
>         >         envelope of that running phase wave with center
>         frequency
>         >         equal to the
>         >         corresponding resonance frequency.
>         >         And the running direction of that phase wave is
>         always from
>         >         base to apex.
>         >         Exactly as Tianying Ren has reported in his then
>         speech making
>         >         paper that I
>         >         have cited:
>         >
>         >         Ren T. (2002) Longitudinal pattern of basilar
>         membrane
>         >         vibration in the
>         >         sensitive cochlea. Proc Nat Acad Sci USA 99:
>         17101-6.
>         >
>         >         The animation of such a phase wave can be seen in:
>         >
>         >
>         http://www.a3ccm-apmas-eakoh.be/aobmm/bm-movement.htm
>         >
>         >         You wrote:
>         >
>         >         **It sounds like you're trying to get away from a
>         >         Helmholtz-like conception
>         >         of resonators or places responding to frequencies,
>         and replace
>         >         it with a
>         >         more time-domain approach that works for a lot of
>         pitch
>         >         phenomena.  But it
>         >
>         >         will work better to put that time-domain mechanisms
>         AFTER the
>         >         what the
>         >         cochlea does.  Each hair cell is a "tap" on the BM,
>         reporting
>         >         a time-domain
>         >         waveform as filtered by the traveling-wave
>         mechanism; that's
>         >         where the
>         >
>         >         pitch-processing nonlinear time-domain operations
>         start...**
>         >
>         >         As you already have indicated in the beginning, you
>         haven’t
>         >         studied the
>         >         booklet entirely. I know for sure that by not
>         studying the
>         >         booklet
>         >         entirely, you have drawn premature conclusions here.
>         >
>         >         It is quite on the contrary. I think that I have
>         explained
>         >         clearly enough
>         >         in the booklet that everywhere along the basilar
>         membrane very
>         >         local
>         >         resonance with a high quality factor takes place.
>         However not
>         >         on the
>         >         primary sound pressure signal, but on the sound
>         energy signal.
>         >         Next to that
>         >         the basilar membrane will react everywhere – but not
>         in a
>         >         resonance mode
>         >         and therefore with much smaller displacements – and
>         will show
>         >         a response on
>         >         other frequency components, including utmost low
>         frequencies
>         >         even until
>         >         stationary pressure signals.
>         >
>         >         And for the explanation of our hearing sense I don’t
>         need a
>         >         time domain
>         >         mechanism at all.
>         >         In the new paradigm, described by me, from all the
>         >         distinguishable
>         >         frequencies next of course to their frequency also
>         their
>         >         individual
>         >         amplitude and phase are transmitted to the auditory
>         cortex.
>         >
>         >         Our brain can directly compare the entire frequency
>         selected
>         >         sound energy
>         >         stimulus with patterns that are stored in our
>         memory.
>         >
>         >         Actually I cannot imagine a much more simpler and
>         faster way.
>         >
>         >         Finally about the definition of Ockham’s Razor –
>         also spelled
>         >         Occam – I
>         >         found on the Internet  the following physics
>         educational
>         >         website:
>         >
>         >
>         http://math.ucr.edu/home/baez/physics/General/occam.html
>         >
>         >         where among others a number of stronger but clear
>         definitions
>         >         are given,
>         >         and I cite:
>         >
>         >         *** If you have two theories that both explain the
>         observed
>         >         facts, then you
>         >         should use the simplest until more evidence comes
>         along.
>         >
>         >         The simplest explanation for some phenomenon is more
>         likely to
>         >         be accurate
>         >         than more complicated explanations.
>         >
>         >         If you have two equally likely solutions to a
>         problem, choose
>         >         the simplest.
>         >
>         >         The explanation requiring the fewest assumptions is
>         most
>         >         likely to be
>         >         correct.
>         >
>         >         . . .or in the only form that takes its own
>         >
>         >         Keep things simple! ***
>         >
>         >         Within this framework I am convinced that I have
>         done my
>         >         utmost best.
>         >
>         >         So I am awaiting for a much better explanation for
>         the
>         >         described beat
>         >         phenomena based on the current hearing paradigm.
>         >
>         >
>         >         Kind regards,
>         >
>         >         Pim Heerens
>         >
>
>
>
>