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Hearing, the cochlea, the frequency domain and Fourier’s series

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In recent weeks, we have seen repeated attempts to suggest that Mathematics is essentially a mind game we make up as an aspect of culture. There has been a very strong resistance to the idea that there are intelligible manifestations of structure and quantity embedded in the fabric of the world (and indeed in that of any possible world). And when test cases have been put on the table, they have been consistently brushed aside as cases where our mathematical modelling has been applied; that is it’s all in our heads.

So, it is appropriate to put on the table a test case that is quite literally in our heads, hearing and particularly how the cochlea works. Video:

We see here how there is a frequency domain transformation that makes use of the mechanical properties of the inner ear. That is, our hearing moves from the time to the frequency domain, sensing pitch; also, subtle timing differences between sound arrivals at our right and left ears help us to locate sound sources in the space around us. As was noted in a comment in the Fourier thread:


KF, 4: >>On how hearing creates a frequency domain transform of sound inputs, driving the onward processing, Wiki is a handy reference:

The stapes (stirrup) ossicle bone of the middle ear transmits vibrations to the fenestra ovalis (oval window) on the outside of the cochlea, which vibrates the perilymph in the vestibular duct (upper chamber of the cochlea). The ossicles are essential for efficient coupling of sound waves into the cochlea, since the cochlea environment is a fluid–membrane system, and it takes more pressure to move sound through fluid–membrane waves than it does through air; a pressure increase is achieved by the area ratio of the tympanic membrane to the oval window, resulting in a pressure gain of about 20× from the original sound wave pressure in air. This gain is a form of impedance matching – to match the soundwave travelling through air to that travelling in the fluid–membrane system . . . .

The perilymph in the vestibular duct and the endolymph in the cochlear duct act mechanically as a single duct, being kept apart only by the very thin Reissner’s membrane. The vibrations of the endolymph in the cochlear duct displace the basilar membrane in a pattern that peaks a distance from the oval window depending upon the soundwave frequency. The organ of Corti vibrates due to outer hair cells further amplifying these vibrations. Inner hair cells are then displaced by the vibrations in the fluid, and depolarise by an influx of K+ via their tip-link-connected channels, and send their signals via neurotransmitter to the primary auditory neurons of the spiral ganglion.

The hair cells in the organ of Corti are tuned to certain sound frequencies by way of their location in the cochlea, due to the degree of stiffness in the basilar membrane.[3] This stiffness is due to, among other things, the thickness and width of the basilar membrane,[4] which along the length of the cochlea is stiffest nearest its beginning at the oval window, where the stapes introduces the vibrations coming from the eardrum. Since its stiffness is high there, it allows only high-frequency vibrations to move the basilar membrane, and thus the hair cells. The farther a wave travels towards the cochlea’s apex (the helicotrema), the less stiff the basilar membrane is; thus lower frequencies travel down the tube, and the less-stiff membrane is moved most easily by them where the reduced stiffness allows: that is, as the basilar membrane gets less and less stiff, waves slow down and it responds better to lower frequencies. In addition, in mammals, the cochlea is coiled, which has been shown to enhance low-frequency vibrations as they travel through the fluid-filled coil.[5] This spatial arrangement of sound reception is referred to as tonotopy . . . . Not only does the cochlea “receive” sound, it generates and amplifies sound when it is healthy. Where the organism needs a mechanism to hear very faint sounds, the cochlea amplifies by the reverse transduction of the OHCs, converting electrical signals back to mechanical in a positive-feedback configuration. The OHCs have a protein motor called prestin on their outer membranes; it generates additional movement that couples back to the fluid–membrane wave. This “active amplifier” is essential in the ear’s ability to amplify weak sounds.[6][7]

The active amplifier also leads to the phenomenon of soundwave vibrations being emitted from the cochlea back into the ear canal through the middle ear (otoacoustic emissions) . . . .

Otoacoustic emissions are due to a wave exiting the cochlea via the oval window, and propagating back through the middle ear to the eardrum, and out the ear canal, where it can be picked up by a microphone. Otoacoustic emissions are important in some types of tests for hearing impairment, since they are present when the cochlea is working well, and less so when it is suffering from loss of OHC activity . . . .

The coiled form of cochlea is unique to mammals. In birds and in other non-mammalian vertebrates, the compartment containing the sensory cells for hearing is occasionally also called “cochlea,” despite not being coiled up. Instead, it forms a blind-ended tube, also called the cochlear duct. This difference apparently evolved in parallel with the differences in frequency range of hearing between mammals and non-mammalian vertebrates. The superior frequency range in mammals is partly due to their unique mechanism of pre-amplification of sound by active cell-body vibrations of outer hair cells. Frequency resolution is, however, not better in mammals than in most lizards and birds, but the upper frequency limit is – sometimes much – higher. Most bird species do not hear above 4–5 kHz, the currently known maximum being ~ 11 kHz in the barn owl. Some marine mammals hear up to 200 kHz. A long coiled compartment, rather than a short and straight one, provides more space for additional octaves of hearing range, and has made possible some of the highly derived behaviors involving mammalian hearing . . .

In short, sinusoidal frequency domain decomposition of sound waves is a key mechanical phenomenon exploited by our hearing system, leading to in effect a frequency domain transformation of the temporal pattern of compressions and rarefactions that we term sound. This is of course closely related to the patterns we explored and discovered using Fourier power series and integral analysis of oscillations and transient pulses.

Where, on the mechanical side, harmonic motion is tied to elastic and inertial behaviour. Which in turn is directly connected to a rotating vector analysis — leading straight to the complex exponential analysis that draws out the full power of complex numbers, form Z = R*e^i*wt, w being circular frequency 2* pi*f (in radians per second), f the cycle per second frequency. All of this ties back to the fundamental frequency cycles and integer-multiple frequency harmonic epicycles in the OP above.

Again, Mathematical study turns out to reflect quantities, structures and linked phenomena which are embedded in the fabric of our world.


PS: Notice, not a few design subtleties?

PPS: The vocal tract, in effect a wind instrument, also exploits fundamentals and harmonics to create auditory, frequency-based patterns as well as transients.>>


We see here yet another case where structure and quantity are embedded in the natural world and are exploited in the design of our bodily organs; here, those for hearing. Thus, literally in our heads. END

22 Replies to “Hearing, the cochlea, the frequency domain and Fourier’s series

  1. 1
    kairosfocus says:

    Hearing, the cochlea, the frequency domain and Fourier’s series

  2. 2
    PaoloV says:

    Very interesting topic and video. Thanks.

    Hearing, Ear Anatomy & Auditory Transduction

    This video follows the path of the sound waves traveling through each part of the ear (outer ear, middle ear, and inner ear), interacting with the tympanic membrane, auditory ossicles, and the bony labyrinth of the cochlea, until it arrives to the hair cells (auditory receptors), located further within the cochlea, that generate nerve impulses in response. -Sound waves enter the ear, passing along the external auditory canal (meatus) to the tympanic membrane (eardrum). The membrane vibrates in response to sound. Low pitch sounds produce low vibration frequencies. Low volume sounds produce low vibration amplitudes. High frequency sounds produce faster vibrations of higher pitch. -The tympanic membrane articulates with the auditory ossicles (three smallest bones in the body, the malleus, incus and stapes). They pass vibrations that initially hit the tympanic membrane to the oval window of the bony labyrinth in the cochlea displacing a fluid called perilymph. The round window at the end of the bony labyrinth facilitates this displacement by allowing the perilymph movement. -The auditory vibrations move in the cochlea (snail-shaped) ascending via the scala vestibuli (scala means stairs) and descending via the scala tympani. Between these two is the cochlear duct, which is filled with endolymph, it is separated from the scala vestibuli by the Reissner’s (vestibular) membrane and from the scala tympani by the basilar membrane. The vibrations ascending the scala vestibuli are transferred to the Reissner’s membrane and the cochlear duct. Here, the hair cells in the organ of Corti, between the basilar membrane and tectorial membrane, receive the vibrations and generate nerve impluses. These impulses are sent to the brain via the cochlear nerve.

    Video Produced by: Brandon Pletsch. “Auditory Transduction (2002).” YouTube. Brandon Pletsch, Aug 26, 2009.

    Music by: “Beethoven: Symphony No. 9 in D Minor, Op. 125 – “Choral”: II. Molto Vivace” by Orchestre Révolutionnaire et Romantique & John Eliot Gardiner

    Video Edited by: V. Kolchenko & Tristan Charran, New York City College of Technology, 2017.

    Here are other videos on the topic:
    Ear and the mechanism of hearing

    Journey of Sound to the Brain

  3. 3
    kairosfocus says:

    PaV, we see here how the dynamics of harmonic motion are used to create a sensor array that responds to frequency, which is much of what we perceive as pitch. Thus, there is an effective mechanical transformation to the frequency domain, embedded in our hearing organs. This is of course coded into our genetic deposit. I gather that the child in the womb is already hearing and processing sound, so that for example familiar voices are responded to at birth (e.g. that of the father). Such is already interesting at the level of the design inference on functionally specific complex organisation and associated information, but also we have a case where — literally in our heads — we see how structure, quantity and linked dynamics are written into the fabric of our world. KF

  4. 4
    hazel says:

    kf writes,

    we have seen repeated attempts to suggest that Mathematics is essentially a mind game we make up as an aspect of culture. …And when test cases have been put on the table, they have been consistently brushed aside as cases where our mathematical modelling has been applied; that is it’s all in our heads.

    What an incredible mischaracterization of what has been said by me and others about math. This is one of the reasons I no longer post here: there is a widespread inability and/or unwillingness to understand any ideas that don’t fit the UD zeitgeist.

  5. 5
    PaoloV says:

    KF @3:

    “I gather that the child in the womb is already hearing and processing sound, so that for example familiar voices are responded to at birth (e.g. that of the father).”

    That’s a very good point.

    Here’s another video on the auditory processing

  6. 6
    PaoloV says:


    Can sound (for example words, music) be “heard” in our mind without having to be received through our ears?

    Here’s another video on ASCENDING AUDITORY PATHWAY

  7. 7
    kairosfocus says:

    H, I disagree with your characterisation and the implication that if I spend much time on it, it will derail this thread. It is fair comment on my part that several objectors have definitely sought to deny that any material aspect of reality embeds substance of structure and quantity that is intelligible, forming a body of core mathematical facts that constrain our reasoning. Things like the implications of a twelve segment rope or a Mobius strip, which has mathematical behaviours independent of our understanding. On your part, while there was some concession that some such facts are facts of the world, there has been a tendency to try to pull focus to the study, cultural conditioning and the like. Think about the apparent subtext in say: “Nice math [as in study of], and nice application [as in, locus is our heads, extending to the world] to the physical world” when in fact the structure, rotating vectors tied to C, was opening up how the frequency domain lurks in the world, something this OP draws out and headlines. Something which, BTW is a matter closely connected to the significance of Fourier, Laplace and Z transforms which were central to much of what I did for years. Indeed for a period I was more in the complex frequency domain than the everyday spacetime one. Before my time, a spectral analysis of economic trends was key to some major work by my Father. Moreover, there has been a clear pattern of objections along the line of the Leff grand sez who, without substantially warranting the that’s what YOU think or believe dismissal. The above is a case of embedding of structure and quantity that is literally in our heads. KF

  8. 8
    kairosfocus says:

    PaV, it seems that subvocalisation is often implicated in perceived voices in the mind, especially while reading. When thinking in a voice, I don’t know. I have heard that some can read a musical script and hear the music in perfect pitch internally. Certainly in his deaf years Beethoven had to rely on that. KF

    PS: Where subvocalisation exists, it can be sampled from muscular movements and long ago I heard of doing the like by putting a mike to the ear — feedback through the circular window of the cochlea.

  9. 9

    The journal “Hearing Health” offers some interesting articles on hearing. Having suffered Meniere’s Disease in the past, it is always interesting to me to find articles about hearing. Take a look at one such article in the Winter 2019 issue, and notice the engineering language used throughout the short synopsis:

    Everyday listening situations require concurrent processes to not only recognize but also apply time-stamps to distinct sound elements while separating them from ongoing background noise. The brain’s decoding of the acoustic scene, known as auditory streaming, includes speech perception. For the brain to make sense of communication sounds, it must recognize distinct acoustic units, such as phonemes and syllables, whose acoustic “boundaries” are perceptible even by infants. This is the difference between hearing, for example, “she got an allergy shot” vs. “she got a scholarship.” The human ability to understand speech is a combination of basic acoustic detection combined with higher-order, phonetic processing that incorporates learning and memory. Richard A. Felix II, Ph.D., a 2016 ERG scientist, and team have been investigating the roles played by “bottom-up,” lower-level sensory processes compared with “top-down,” higher-level cognition. The team has been using a mouse model to glean insights into auditory processing mechanisms common to all mammals and that underpin the acquisition of language. The onset of sound is a cue that alerts the brain to new, incoming acoustic input. The auditory system is highly sensitive to abrupt or loud or high-pitched sound components, specifically through “octopus cells” (with tentacle-like dendrites) in the brainstem’s cochlear nucleus. Octopus cells have been shown to have a precise time-locked firing (spiking) response to the onset of broadband sound stimuli. These cells target the auditory brainstem region called the superior paraolivary nucleus (SPON), which has been shown to be involved in the processing of rhythmic sounds. The SPON demonstrates a sound-triggered onset spiking response, sometimes alongside an offset spiking response. It is thought that this onset/offset combination helps represent natural sounds such as vocalizations. Testing neural reactions using a variety of sound frequencies and intensities, including mouse calls, Felix and colleagues detail the role of the SPON in extracting vocal communication information. In a paper published in the European Journal of Neuroscience on July 18, 2018, the team’s study helps explain the onset and offset spiking responses of these auditory brainstem neurons in terms of decoding natural sounds. The researchers show that the broadly tuned, onset spike may improve the signal-to-noise ratio to aid in understanding a subsequent, frequency-speci?c peak. As a result, the SPON may provide a dual inhibition mechanism that allows the tracking of the starts and ends of phonetic units in speech in order to send accurate speech-sound signals bottom up to the brain.

    That the audio/visual gift most of us have is a design/engineering marvel is beyond dispute. And yet folks like Richard Dawkins milk multiple millions in wealth from the Darwinian myth is a monumental fraud.

  10. 10
    PaoloV says:


    Check this out:

    Actors Tom Hulce and F. Murray Abraham discuss the famed dictation scene from Amadeus, the movie.

    Mozart heard the complex music in his mind before any mortal being could ever hear it?
    Can somebody explain it? (obviously skipping the artistic freedom of the film director).

  11. 11
    PaoloV says:

    Ayearningforpublius @9:

    Excellent comment. Thanks.

  12. 12
    kairosfocus says:

    PaV, I suspect that deep practice and skill associates music scores with internal hearing just as much as external sound with the textual representation. I recall doing a sound experiment with a lab partner who could catch the point where the resonance caught musical notes. He did so perfectly, better than I could get by IIRC trying to get physical beats between neighbouring sounds to die out — tuning fork vs tube. Piano tuners often work by ear too. But, truth be told I am utterly unmusical. KF

  13. 13
    kairosfocus says:

    F/N: I should add that a prism’s dispersion of light through frequency/wavelength-sensitive refractive index and the similar effect of a diffraction grating also transform physically from time to frequency. In a precisely quantitative structural pattern spread across space, yielding the familiar rainbow spectral pattern. KF

  14. 14
    PaoloV says:

    KF @12:
    “I suspect that deep practice and skill associates music scores with internal hearing just as much as external sound with the textual representation.”
    How does that association work? What mechanism could have preceded this? How could such a mechanism evolve?

  15. 15
    kairosfocus says:

    PaV, in part, our brains have neural networks that are capable of forming differing degrees of coupling between stages in response to trial and success. The judging of degree of success itself being another challenge. There will be all sorts of onward issues but it boils down to, the structures and quantities– as well as dynamics of harmonic motion [thus differential equation patterns] are there as part of the framework of the world. Our ears are essentially mammalian, and we developed (or was it, were gifted with?) speech as a means of communication. I recall, a son who as a toddler was seen moving in rhythm, then we realised, he was responding to windshield wiper blades. And, more. Being a case of functionally specific, complex organisation and/or associated information, we have every good reason — I no longer have reason to take ideologically driven closed minded objectors seriously — to hold that the origin is not by blind watchmaker incrementalism, but by design. But that holds across the tree of life from unicellular forms to us. We are a new level, highly intelligent creative, morally governed, designing life. KF

  16. 16
    PeterA says:

    “I recall, a son who as a toddler was seen moving in rhythm, then we realised, he was responding to windshield wiper blades. ”
    That’s really funny. Kids are delightful. Too bad we grow up to turn into boring stressed out creatures devoid of curiosity.

  17. 17
    daveS says:


    we have seen repeated attempts to suggest that Mathematics is essentially a mind game we make up as an aspect of culture. …And when test cases have been put on the table, they have been consistently brushed aside as cases where our mathematical modelling has been applied; that is it’s all in our heads.

    I haven’t been checking in much lately, but it appears I’ve missed quite a bit! I wonder who this scalawag is? 🤔

  18. 18
    kairosfocus says:

    DS, no one individual — barring some candidates to be sock puppets. Note, too, a trend (as opposed to a formal admission). One of the challenges I have been giving is, construct three loops of paper, a, b, c. Let A be an ordinary loop and b and c be Mobius strips with one half turn twist (for consistency, anticlockwise). Snip arounf the loop, in the middle for a and b. For c, snip at 1/3 the way across. Observe the sharp difference in the three cases. a: two narrower simple loops. b: a double-length loop with IIRC four half-turn twists. c: a one twist mobius loop like a pared down version of the original, interlocked with a longer, multiple twist, narrower loop. These results are objectively observable, reproducible and independent of how we may think about relevant axiom systems or whether or not we understand why they happen. They manifest one way in which rational, intelligible principles of structure and quantity — here, spatial ones — are embedded in our world and in bodies in it. The m-strip of course exerted a shaping influence on how the domain of Math known as topology, was framed. At less complex levels, the Egyptians long since used twelve-segment ropes to specify right angles through the 3-4-5 relationship and a triangle formed on the diameter of a semicircle with third vertex along the arc must be a right angle triangle. Other things relate to rates and accumulations of change (I am presently pondering Rahm and Laffer curves in macro regarding government size) and the OP illustrates how transform to the frequency domain is literally built into our heads, not just Fourier’s mathematical calculations. Where, on distinct identity of a possible world, we immediately detect nullity, unity (including complex unity) and duality, thus per von Neumann, N, from which Q, R and C follow with much more in tow. Sqrt(-1) turns out to have natural roots in vector rotation, an important natural phenomenon. And more. The overall point is that Mathematics has a dual character, the (study of the) logic of structure and quantity. The substance in key part is embedded in the fabric of this or even for some aspects any possible world, and our culturally influenced study is thus subtly but strongly shaped by such embedded aspects. One problem has been dismissiveness towards logical-mathematical demonstration on first principles such as distinct identity without provision of any significant counter-warrant. Another has been resort to nominalistic views, which turn out to be self referentially, irretrievably flawed, e.g. one cannot state the nominalistic case without implying reality of key relevant abstracta. Where, further, things like first principles of right reason (LoI, LNC, LEM) are inescapably so. KF

  19. 19

    I am incredibly fascinated by the video presented in this article. After playing it several times, I then recall the times sitting in a concert hall listening to some incredible music with instruments ranging from the kettle drums, to the horns, to the woodwinds, to the strings and more. A choral arrangement is an added bonus.

    Then I realized that I have an orchestra of my own … right inside my head. And I didn’t even have to rehearse and practice. Right there in my own head. And yours as well.

    I wonder if anyone has put such visuals together with an orchestra for a full performance of one of the great masters? Does anyone know of such a work? I think I could sit still for hours soaking it all in.

    Just think … a full orchestra, right in my own head.

  20. 20
    daveS says:

    DS, no one individual — barring some candidates to be sock puppets.

    I’ll keep my eyes peeled then.

  21. 21
    ET says:

    And to think that blind and mindless processes produced our auditory system. “Parts came from the reptilian jaw, I tell you. Nature is more clever than you think!”

    And it didn’t need no steekin’ maths, neither. 🙄

  22. 22
    kairosfocus says:

    ET, I know a finely engineered system when I see it, and the cochlea is a masterpiece. I don’t even want to think about how it assembles itself during gestation — another dimension of the complex functionally specific organisation of living things. And on this I no longer take the common ideologically loaded objections that pivot on appeals to endless statistical miracles seriously. Where BTW, it is not irrelevant to note that the growing baby already recognises and processes sounds, i.e. it is learning. I here think of cases where newborns react to parents’ voices and also of an old trick of using an old fashioned ticking clock to soothe as it mimics mom’s heartbeat. Learning is credibly happening in utero, a sign of intelligent behaviour. That text in Lk where John in the 6th month responds to Jesus (likely the 1st) is in some ways unsurprising. KF

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