How We Hear (or, An Introduction to Fourier Analysis)

Monica Headphones

Why, yes, I do Groot-dance when nobody’s around.

(Thanks to my dad for suggesting this topic! It’s perfect—scientific, interesting, beautiful, and such a wonderful and wonder-full part of our experience of life.)

Generally speaking, our sensory organs are things of wonder. Not to say that we don’t understand what’s going on with them—increasingly, we do—but that doesn’t make them any less marvelous. Our skin, our eyes, our noses, our tongues, all deserve their own posts. Today’s is about our ears.

If you only remember a few interesting tidbits about human anatomy, one of them might be that the middle ear (located, oddly enough, between the inner ear and outer ear) contains the three smallest bones in the human body—the hammer and anvil, which fit thematically, and the stirrup, which doesn’t really fit the naming theme but definitely looks like a stirrup. These three bones connect the eardrum or tympanum, which sits at the inner end of the ear canal you’re not supposed to stick Q-tips into, to the cochlea, which is responsible for the signaling that affects your sense of balance and communicates sound to the brain.

Ear Structure.png

Simplified diagram of the outer, middle, and inner ear.

To get to this deeper area of the ear, sound waves first have to make it to our outer ears, where the dish-shaped external part of the ear (or pinna) helps bounce the waves into the ear canal. The sound waves then travel as far as the eardrum, which is exactly what it sounds like (so to speak)—a tightly-stretched little membrane that vibrates in response to the waves that hit it.

Here’s where some amazing math ties in with our amazing biology. The eardrum communicates these sound waves with its vibrations, which are translated through the tiny ear bones into the inner ear. Once the vibrations hit the spiral of the cochlea, they move through the liquid inside it, affecting the basilar membrane that lies along the length of the inside of the cochlea. The membrane is very stiff and thin at the outer edge or base of the cochlea and becomes wider and more relaxed as it spirals inward. This difference in size and stiffness means that different parts of the basilar membrane vibrate in response to different frequencies, just like tuning guitar strings more tightly makes them vibrate at higher pitches.

That probably sounds kind of cool, but not very mathematical, so let’s consider how it works in real life. We almost never hear just one sound at a time—think about how many different sources of noise are around you at any given moment. Even if we’re used to tuning them out, on a physical level all those sound waves from different sources, with different directions and pitches and patterns, are all making their way into our ear canals. It’s a wonder we can make sense of any of it, really.

This is why the basilar membrane is so fantastic. Thanks to its structure, it actually breaks down the muddle of sound waves into specific frequencies that touch off vibrations in precise parts of the membrane. The specifics of the human cochlea mean that we hear sounds in a general range from 20 Hz (vibrations per second) at the wide, relaxed end to 20,000 Hz at the stiff, narrow end.

Essentially, what’s going on inside our ears is a complicated piece of mathematics called a Fourier transform. Starting with a mishmash of a waveform that’s a combination of different waves, Fourier analysis identifies contributions at different frequencies, allowing us to reconstruct the individual signals that went into it. Here’s a (very inaccurately drawn) example of what that might look like for the multiple sound waves that assault us from every direction.

Fourier Transform.png

Breakdown of sound profile for your average afternoon out in the park. Science has yet to explain why your own kid will ALWAYS be louder than everyone else’s at key moments.

This brings up another aspect of hearing, the directional component. This goes back to our external ears, the pinnae—their shape causes differences in sound waves bounced from different angles, so we can determine the vertical location of the sound source. Having a matched set of ears makes it possible to find the horizontal direction of the source, based on the difference between the sound waves that make it to each ear.

Once the sound waves make it to the basilar membrane, they’re translated into electrical impulses by stereocilia, parts of the tiny hair cells along the membrane. These impulses are what get passed on to the nervous system, with the information already broken down by frequency and directional cues.

I love the idea that we’ve only fully understood the theory of Fourier transforms for about 200 years, but tiny membranes and hairs inside our ears have been doing it all along. It’s an elegant way of making order out of what appears to be a mess, and understanding what goes on all the time in my ears makes me marvel even more at what our bodies do constantly without our awareness. There’s always more to learn!

If you’d like to learn more about Fourier transforms and analysis, you can find some good explanations out there that don’t start out by slapping you in the face with all the math. There are also other resources specifically about Fourier transforms in hearing, like this blog post that gets much more into the mathematics but retains the sense of wonder. Enjoy your resEARch! (Yeah, I’m done now.)

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2 thoughts on “How We Hear (or, An Introduction to Fourier Analysis)

  1. How cool! I know that you listen to me; it’s nice to have it confirmed in a post. 😁

    The material about the process in the cochlea is new to me, probably because I learned about the Fourier transform in a math class under a prof who probably had no idea of the finer points of human hearing. The deeper you go, the more amazing it all gets.

    On Wed, May 10, 2017 at 12:17 PM, Something of the Marvelous wrote:

    > SomethingMarvelous posted: ” (Thanks to my dad for suggesting this topic! > It’s perfect—scientific, interesting, beautiful, and such a wonderful and > wonder-full part of our experience of life.) Generally speaking, our > sensory organs are things of wonder. Not to say that we don’t un” >

    Like

  2. Pingback: Sliding Scale: The History of Pitch | Something of the Marvelous

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