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, an entire piece. When combined with amplitude (or the power or height of sound) and graphed they produce a wave and that wave tells you the frequency and its height tells you the amplitude.

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), and 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 as the frequency goes up, the power of sound goes down. This pleasing rise and fall is called pink noise, and we like it. It hits the sweet spot: not random, not too predictable, it is postualted that as the natural world is orientated to order, and a fractal shaped order at that, we are predisposed to like and write pink noise music.

On the other hand, if as frequency goes up and the power of the sound also goes up, we call it brown noise, and it 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 just wanders, unpredictable.

Our other tools in this series (add link) talk about the Hurst value. Hurst and the spectral exponent are linked by this calculation: the spectral exponent equals 2 times the Hurst value minus 1. For this reason Hurst falls between 0 and 1, with a Hurst of 0.5 being white noise, 0.75 being pink, and 1.0 being brown. What we found pleasant in our other tools falls into the pink realm.

It should be noted that the calculation is only approximate, and holds true for fractional Brownian motion. In real life, music is mixed and contaminated with other noise values—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 goes between a spectral exponent of 0 and 2, as you drag toward either end, you can hear the noise change character.

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 way he does this is by selecting notes that naturally 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, in a study that took almost 2000 selections of Western music, the vast majority of them conformed to the 1/f arrangement of notes, pink noise music is the default order.

So coming back to our fractal correlation, a fractal has the same shape no matter the level of zoom,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’s the same note held too long, no interesting variation between scales. Pink noise sits between: structured and self-similar at every level.

Interestingly, because 1/f is predictable and easy to generate mathematically, it is possible to generate music that follows this rule.

Musicians

Richard Voss, like the mathematician Benoit Mandelbrot (who himself worked with fractals), used IBM’s 1970s mainframe computer to show that much of music across cultures and time periods follows 1/f or pink noise rules. He took a varied sample of music and even speech and showed that the vast majority of them followed the 1/f rule. Further research has confirmed his findings, and set researchers and musicians in search of pink noise music.

This theory opened up a new branch of music experimentation.

Some artists who worked with pink noise are Brian Eno, Iannis Xenakis, Ligeti, Pärt, and Gary Nelson. The most famous of fractal music is Gary Lee Nelson’s Fractal Mountains,