Quora user Marc Ettlinger recently sent me a paper by Sherri Novis-Livengood, Richard White, and Patrick CM Wong entitled Fractal complexity (1/f power law) determines the stability of music perception, emotion, and memory in a repeated exposure paradigm. (The paper isn’t on the open web, but here’s a poster-length version.) The authors think that fractals explain our music preferences. Specifically, they find that note durations, pitch intervals, phrase lengths and other quantifiable musical parameters tend to follow a power law distribution. Power-law distributions have the nifty property of scale invariance, meaning that patterns in such entities resemble themselves at different scales. Music is full of fractals, and the more fractal-filled it is, the more we like it.
The image above is part of the famous Mandelbrot set. Here’s an article explaining what it is and why it’s so cool. Fractals are all around us: in clouds, trees, mountains, coastlines, and our own bodies. Basically everything we enjoy looking at has a fractal quality. So it’s no big surprise that the same should be true of things we enjoy listening to as well.
It’s easy to grasp what a fractal is by looking at pictures like the one above. It’s not so easy to imagine what the musical analog would be. You can think of a musical fractal as groups of events combining to form larger groups of events, which themselves combine still larger groups — loops within loops within loops. Each group reflects the form of the whole, in pitch contour or patterns of tension and resolution or whatever.
So what does the 1/f in “1/f noise” refer to?
Here’s some background from Scholarpedia:
1/f noise refers to the phenomenon of the spectral density, S(f), of a stochastic process, having the form
where f is frequency, on an interval bounded away from both zero and infinity.
You might know 1/f noise by its other name, pink noise. It’s an intermediate between white noise and random walk (Brownian motion) noise. White noise has no correlation between elements in time, while Brownian motion has no correlation between increments.
Fun calculus fact: Brownian motion is the integral of white noise. However, there aren’t any simple, even linear differential equations that generate pink noise. The widespread occurrence of pink noise in nature suggests that a generic mathematical explanation might exist, but no one yet knows what that might be.
Examples of fractals and 1/f noise in biology
- The time series made up of intervals between successive EKG peaks of the human heart.
- The postural sway of a person standing on a platform.
- Fluctuations in the time density of action potentials, and the activity of ensembles of neurons in animal brains generally.
- The structure of blood vessels and other body transport systems.
Examples of fractals and 1/f noise in music
- The power spectrum of intensity fluctuations in a recording of Bach’s Brandenburg Concerto Number 1. Pitch and amplitude fluctuations in a wide variety of other music recordings.
- Power spectra found in variation of rhythms, including walking and finger-tapping with a metronome.
- Vocal processes related to the production of music, such as aspiration patterns and amplitude modulation.
- The Amen break, which is full of golden ratios, a specific form of self-similarity.
- The structure of pieces by Stravinsky and Cage.
A concrete example of fractal music: self-similarity in the blues
I tried to think of a good example of music with a fractal structure, and pretty quickly hit on the twelve-bar blues. The blues is all about patterns of three versus four at different scales and along different axes.
- The shuffle beat divides 4/4 time into eighth note triplets.
- Quarter note triplets, hemiola and three-against-four polyrhythm appear frequently in blues.
- The standard blues song form is three groups of four bars each.
- Both the I-IV-V chord progression and the pentatonic scale can be organized around the circle of fifths/fourths. A perfect fourth is two frequencies with the ratio 4:3. A perfect fifth is the inverse, 3:4.
- Blues harmony is based around the contrast between major thirds in the chords (four semitones) and minor thirds in the melody (three semitones.)
- In the V7 chord, you also find natural seven (four semitones above the fifth) versus flat seven in the melody (three semitones above the fifth.)
- When you go from I to I7 or from IV to IV7, you’re changing from a three-note triad to a four-note seventh chord.
- Diminished chords figure heavily in blues; they consist of four groups of three semitones each.
Blues also has some other more general self-similarity: each four-bar phrase consists of a call and response, and the whole form consists of two call phrases answered by the response phrase.
So why do we like fractals in the first place?
Since our brains naturally display fractal properties in their ordinary operations, it shouldn’t be surprising that we find fractals interesting. Livengood, White and Wong point out that “[o]ptimal fractal complexity, which can be thought of in terms of a ratio of unpredictability-to-predictability, elicits peak mood and peak memory performance.” But why do we find this pleasureable? Is it just a meaningless coincidence that we enjoy seeing our our mental processes reflected back at us? Maybe. But maybe there’s something more evolutionarily meaningful at work. In fact, fractals may be crucial to our well-being. Vijay Sharma explains:
When fractal complexity [in heart rate] breaks down, the breakdown can reflect a cascading of the system into true randomness or a reversion into periodic order. Heart failure may cascade the system in either direction, producing random or periodic behavior; in both cases the fractal complexity is seen to decrease, and in both cases the effect is associated with increased mortality. Fractal complexity and chaotic behavior of the heart have both been found to decrease with ageing, and this loss of complexity is also believed to be detrimental.
Why is the loss of chaotic behavior harmful? Loss of chaotic behavior clearly creates a loss of flexibility in the system. However, it also leads to a loss of information storage and generation. The ability to store and transmit information is lost because random behavior is meaningless, and periodic behavior simply repeats the same information over and over again. This is reflected in the loss of fractal complexity in the heart rate signal; the ‘memory effect’ conferred by the fractal structure is lost. Chaotic behavior is unpredictable behavior, and unpredictable behavior allows for a physiological ‘freedom of expression’; the key to generating useful information is the ability to change.
Sometimes people read these kinds of mathematical analyses of music and accuse me of being too intellectual or unfeeling. For me, though, the appreciation of mathematical beauty is inseparable from the appreciation of any other kind of beauty. What’s the difference between the gratification you get from hearing music and the kind you get from seeing a tree or a coastline? I don’t think there is one.