Hearing the Math of Vibrations

In about 0.1s the pressure oscillated 31 times, corresponding to a frequency of 31/0.1s = 310Hz. As this pressure wave hit my ear, it jiggled my ear drum which is connected to a spiral shaped tube of fluid in my cochlea which contains a long structure made of several membranes, hair cells and nerves called the organ of Corti. The critical property of this structure which allows our ear to “hear the math of vibrations” is that it is tapered, thick and stiff at one end, thin and flimsy at the other. 

While the entire organ of Corti gets jiggled by this changing pressure as a function of time, most of it vibrates and flexes out of time with the oscillating pressure and not much motion results. There is one and only one little slice of the organ which when jiggled will naturally oscillate 310 times per second. When this one slice gets jiggled at 310Hz, it will jiggle more and more as each oscillation is perfectly timed to push it in the same direction it is already going. This mechanically amplifies the motion sufficiently for the organ of Corti to respond and alert the brain, “hey the 310Hz section of my Corti is being stimulated!” 

Most people do not have perfect pitch, which means their brain doesn’t actually know which specific section of their Corti corresponds to 310Hz. This is where the math gets really cool. No sound waves are “perfect sine waves” meaning a pure tone of 310Hz in this example. Even if you attempt to produce a perfect sine wave, those sound waves bounce around in your ear canal and the pressure wave that reaches your Corti is more complicated. To see what is going on, we can use math to do what your Corti does mechanically! A Fourier transform converts a function of pressure vs time into a function of loudness vs frequency, also known as the power spectrum. I want to emphasize how cool this is: The fourier transform is an infinite dimensional linear transformation and your ear does it purely mechanically! Biology has been using this abstract mathematical operation for millions of years, but we only discovered it in 1822. It now underpins all of modern physics and huge swaths of mathematics.

This is what your organ of will Corti “hear” when my recorded boop is played back. You can literally think of the x-axis as labeling locations along your Corti; the height of the graph is the amount of nerve stimulation those locations receive due to their mechanical stiffnesses and corresponding resonant frequencies. 

There is quite a lot of information here! Rather than transmit all of this to your brain, your ear’s nervous system actually summarizes this information, sending pulses to the brain which convey the relative locations of the spikes. The more complex the pattern of spikes, the more complex the pattern of nerve pulses becomes. For a more detailed description of the biology and neuroscience with some beautiful diagrams, have a look here.

You can see that most of the power of my boop is right at 310Hz but there is quite a lot of power at 620Hz, 930Hz, 1240Hz, 1550Hz and 1860Hz. You’ll notice that these spikes are 2, 3, 4, 5 and 6 times the base frequency of 310Hz. This is no coincidence! Any time you oscillate anything to make sound – a column of air in your throat or a string on a guitar – that physical system can also support oscillations at integer multiples of the base frequency. This is because these higher frequency oscillations also satisfy the physical boundary conditions of the base frequency; for a string, the fixed points at the ends, for an air column, the pressure at the open end and the displacement at the closed end. 

Some of the energy carried in the base frequency makes its way to these higher “modes” of oscillation, known has harmonics due to physical interactions which are out of scope for this discussion (if you are interested, this effect is due to non-linear coupling between modes and “inharmonicity”). 

If I record a boop at twice the base frequency, 620Hz, this is the corresponding power spectrum:

In this case most of the volume is at 620Hz with spikes at 1240 and 1860. This means that if I play these two boops together the physical locations of the spikes of stimulus on my Corti overlap! 

These two boops played together require less information to be conveyed to the brain than would otherwise be required for boops that were not simply related by a factor of 2 in their base frequency. If I boop at 330Hz, the spikes do not nicely overlap with those of the 310Hz boop:

This requires more information to be communicated to the brain. Most people will interpret these boops played together as somewhat unpleasant or “crunchy” sounding. In context with music this sound can convey complex emotions with degrees of beauty, sadness, apprehension, etc. These associations are cultural and are learned. The math behind the relative complexity or simplicity of spectral information when notes are played together is universal.