How Pink Noise Music Follows a Fractal Pattern

Pink Noise

But, you ask, how can music be a fractal?

A good question, and one that seems odd on the surface. Of course, music does not have an infinite number of notes, nor does it go on to infinity; that is true, but what we do say is that music moves in a fractal-like fashion, and pink noise music in particular meets the criteria.

Like a fractal, music has measures that are self-similar (they look like each other), and like fractals, they repeat, and even mirror each other at a “distance” of similarity; they keep a memory in their beat.

Frequency, Amplitude, and Power

But we are getting ahead of ourselves when we should be looking at the beginning of the idea. Music can be defined as an intersection of frequency and amplitude.

When we talk about frequency in music, we don’t mean high notes versus low notes; we mean how often things change, at whatever point you look at them, at the scale of a note, a phrase, or an entire piece. When combined with amplitude (the height of the sound) and graphed, they produce a wave: how often that wave repeats is the frequency, and how tall it stands is the amplitude. Power is amplitude squared, and it is power we track against frequency from here on.

The relationship between frequency and power can be captured in a single number (calculated on a log scale) called the spectral exponent, written α (alpha). For the noises we are concerned with here, its value falls between 0 and 2.

We classify the results like this:

• α = 0: power doesn’t fall at all as frequency rises — white noise
• α = 1: power falls in perfect proportion to frequency rising — pink noise
• α = 2: power falls steeply, low frequencies dominate — brown noise

In practical music terms, this means that as the frequency goes up, the power of sound goes down in exact proportion. This balanced trade-off is called pink noise, and we like it. It hits the sweet spot: not random, not too predictable. It is postulated that because the natural world is oriented to order, and a fractal-shaped order at that, we are predisposed to like and write pink noise music.

On the other hand, if the frequency goes up and the power of the sound also goes up, we call it blue noise; its exponent is negative, which is why it sits off the end of our scale. Brown noise, where power falls away faster than pink, is muddy and displeasing to the ear; too much bass, too much repetition, boring, and predictable.

If noise is just random, we call it white noise: it scatters, with no memory from one moment to the next.

Our other tools in this series talk about the Hurst value. Hurst and the spectral exponent are linked, but the link comes in two pieces. For fractional Gaussian noise, the family our text and birdsong tools measure, the spectral exponent equals 2 times the Hurst value minus 1. Hurst lies between 0 and 1, so a Hurst of 0.5 gives α = 0, white noise, and a Hurst of 1.0 gives α = 1, pink. Brown noise sits outside that family. It is the running sum of fractional Gaussian noise, called fractional Brownian motion, and there the spectral exponent equals 2 times Hurst plus 1, so ordinary Brownian motion at Hurst 0.5 lands at α = 2. What we found pleasant in our other tools leans toward the pink end.

It should be noted that these formulas are exact only for idealized fractional noise. Real music is mixed with other noise sources—you can see this in the two Hurst values our Bird Song tool returns.

Of the three, pink is the one that sounds good to us. Give it a try in our tool down below. Our noise generator lets you hear the difference between white, pink, and brown noise. The slider ranges from a spectral exponent of 0 to 2. As you drag toward either end, you can hear the noise change character, and the plot underneath shows the same thing to your eyes: at 0 the notes scatter, at 1 they drift and return, at 2 they crawl.

What Sounds Pleasing

We are intrinsically wired, it seems, to hear these patterns, and we pick them up even if they are presented in a subtle way.

Bach, for example, often iterates his compositions (repeats a theme throughout his piece, returning like an overall pattern), and the result is that his note-to-note fluctuations fall into a 1/frequency (1/f) pattern. He also makes his variations align with each other, so they nest, like nesting dolls, symmetrically ordered. We hear this as pleasing to the ear. The mind likes order, and 1/f order in particular. In fact, a study of 1,788 movements from 558 Western classical compositions found that the overwhelming majority obeyed a 1/f power law, this time in their rhythm, the spacing of the notes in time rather than their pitch. Pitch had already been shown to behave this way; what the study added was that the same order governs the clock as well as the tune, and that individual composers sit at reliably different exponents.

So coming back to our fractal correlation: a fractal has the same shape no matter the level of zoom, and 1/f noise has the same statistical structure at every scale. If you zoom into a 1/f signal, looking at it in small units of say one second instead of one minute, the pattern looks the same.

White noise has no structure at any scale (it’s just scatter). Brown noise has structure, but it is all slow drift, the same neighborhood held too long, with little variation between scales. Pink noise sits between structured and self-similar at every level.

Interestingly, because 1/f is cheap to generate mathematically, it is possible to compose music that follows this rule by machine.

Musicians

Richard Voss and John Clarke used 1970s mainframe computers to show that much of music across cultures and time periods follows 1/f or pink noise rules. They took a varied sample of music, drawn from radio broadcasts of several genres, and even speech, and showed that the vast majority followed the 1/f rule. Voss later worked at IBM’s Watson Research Center alongside Benoit Mandelbrot, who coined the word fractal. Further research has confirmed their findings and set researchers and musicians in search of pink noise music.

This theory opened up a new branch of music experimentation.

Voss and Clarke composed directly from 1/f sequences themselves, and Charles Dodge and Gary Lee Nelson followed. Others, including Iannis Xenakis, György Ligeti, and Brian Eno, worked with stochastic, fractal, or generative procedures more broadly, without necessarily targeting a 1/f spectrum. The most famous piece of fractal music is Gary Lee Nelson’s Fractal Mountains, which you can hear below.