How Your Ears Do Math Better Than Mathematicians
When you listen to music, when its waves of sound collide with your ears, you don't hear a wall of sound. A great deal of information might travel in a sound wave and, if that sound wave were actually a giant wave of water rushing onto a beach, you might expect to feel it as a big shove like any other big wave of water coming in from the ocean.
Except that's not what happens when this particular wave hits you: Standing there ankle deep in the surf, you brace for it to crash against your body, but when it does arrive, it's not a "hit" at all. Instead, you feel a hundred different things at once, all on different parts of your body. Some places it's a cool brushing. Some places it's soft slap, or the feeling of a light sunburn. And then the wave has passed.
Of course, that's not how waves of water work. But it is how sound waves work, at least when they're introduced to a human ear. Instead of a wall of noise, your ear percieves a variety of different frequencies. Those frequencies might add up to still be "noise," but they might also be speech or they might be music. It's actually a mystery how this translation happens, the exact method by which we turn the big, looming wave into differentiated ripples.
The assumption was always that ears use something akin to a Fourier transformation. A Fourier transform, named after the French mathematician who also identified the Greenhouse Effect, is essentially when a sound wave is stretched way out until its details are revealed. In more mathy terms, you take a signal, which is a mathematical function of time—a mechanical thing of air molecules traveling through space—and turn it into an array, or a series of different frequencies. The Fourier transform is found all over science, and not just sound.
The transformation is done through what's called an "integration" of the original, mechanical function of time. (If you've taken calculus, you should remember integration.) Basically, this is taking that function and recovering information from it by mathematically slicing it up into tiny bits. It's pretty neat. This, it turns out, is how we get meaning (words, music, whatever) from sound (that big wave in the ocean). Or so scientists have thought.