Physicists Use a Leaky Faucet to Find Chaos in Music

​A pair of physicists has devised an experiment in which true "natural" chaos is adapted into music. Using the random (yet mediated) dripping of water from a faucet, the team was able to demonstrate experimentally an emerging idea about how patterns and noise relate to aesthetic preferences, confirming that most music, from classical to rock, is composed with some sense or degree of sonic disorder. Noise.

The tightest rhythm, the most cliched melody, the 3'30" verse-chorus-verse—it was carved from chaos, and within it, chaos might still be found. This is what the mathematics of music demonstrate: Both the melodies and rhythms of a composition obey power laws, relationships that give us the probability of this or that pattern occurring with respect to the previous event. If I play a "c," how far away (and how unrelated) might we expect the next note to be?

​A simple experiment devised by physicists Nitica Sakharwade and Sayak Dasgupta, both of the Indian Institute of Technology, provides an illustration as to how music might correlate to noise, or different varieties of noise. It's based on a leaky faucet, a particularly well-suited, if subtle, source of chaos.

First, understand that there are different sorts of noise. White noise is what we usually think of. This is where successive values of some sound (or whatever) have no relationship to the previous values. If one value is 5, the odds are just as good that the next could be 1,002 as it could be 6 or 4 (assuming a big enough range).

And then there is ​Brownian noise, which is a much more sensical sort of randomness. While a white noise particle might jump from any point to any point within a given range, a particle experiencing Brownian noise acts as though it's being shoved around by a bunch of other particles who are all pushing in random directions and with random magnitudes. So, instead of jumping all over the place, a Brownian particle just kind of staggers around. This is realistic randomness.

If you were to look at the rhythm or pitch frequencies in music (patterns, generally), they would be somewhere in between white noise and brown noise. It's been shown previously that music, most any sort, obeys a power law but there are different varieties of power laws corresponding to different sorts of noise. And by adjusting the flow rate from a faucet (here a tube from one bucket to another), the frequency with which droplets fall can be made more or less chaotic.

Here's what the actual power law looks like:

Pretty simple, eh? The main thing is the β exponent, which, when it equals 0, means that we can expect perfect white noise. As β changes, the fraction takes on very small values very quickly. This is the general definition of a power law: an intense clustering of probabilities around one narrow range, with those probabilities falling off very quickly as we get even just a tiny bit further away.

Graphically:

The Indian researchers, who posted their findings last week to arXiv, recorded drops as they collided with a metal bowl located underneath their "faucet." These collisions were recorded as data points on a spreadsheet which, over time, revealed droplet frequencies.

"This is where our project gets musical," Sakharwade and Dasgupta write. "To find out regimes which are chaotic we turn the drop time data into frequencies by taking the time differences between successive drops and using f = 1/(δt), and scaling it up to audible frequency range. We then adjust these frequencies to pentatonic major note scale in C Sharp, which enables us to relate the frequencies with our intuitive understanding of music."

And so we have the automatic music created via varying grades of chaos, courtesy of a sink and GarageBand (and MatLab).

"In the next part we listened to the tracks thus produced and chose three tracks corresponding to three flow rates which characterize the totally predictable, chaotic and random noise regimes," the physicists explain. "We then do a power spectral analysis of the data obtained to get the β parameter after a linear fit to the data points."

The point of the study, aside from a neat demonstration of chaos becoming music, is to verify that the power law above does in fact hold when compared to actual chaos (the faucet set-up). "The range of β suggests that human music keeps a balance between predictability (β = 2) and randomness (β = 0)," the paper concludes. So: Patterns alone don't make music, it takes noise as well.

Makes sense. Isn't the opposite of boring noise?