Surprise me - But not too much

Flute recording. Source.

A decades old paper made me philosophical.

In 1975, Voss and Clarke, two physicist from Berkeley, studied noise in electronic devices. For the fun of it, they also analyzed the spectra of different types of music. They found that the fluctuations in loudness and pitch decrease with the inverse of the frequency; they have a 1/f spectrum. This finding was basically independent of the type of music; Western, Oriental, Blues, Jazz, Classic all showed the same pattern. Voss and Clarke’s musical power spectrum made it into Nature.

A Fourier-transformation of a 1/f power spectrum leads to a power-law decay in the autocorrelation time of the fluctuations, meaning there are correlations over all times, rather than over a characteristic decay time as is most often the case. Physicists like power-law autocorrelated fluctuations because systems show them on a critical point, ie if something cool is going on, you get a power law. The opposite is not necessarily true though; there are more mundane ways to get a power law, but that hasn’t deterred enthusiasts. The 1/f spectrum is scale-invariant, so it has – theoretically – no preferred time scale or frequency to it, as one might have expected to be present in music.

In the 70s and 80s everything power-law was chique, and not all of that power-law-finding was very meaningful. To some approximation, in some parameter range, everything is a power-law. If you put your data on a log-log plot, you can make a linear fit, at least over some range. Yet, strictly speaking nothing really is a power law. And of course music doesn’t really have a 1/f spectrum either. To begin with, because it doesn’t use the full frequency spectrum, most of which we couldn’t hear. Also, Mozart hasn’t been composing since the Big Bang.

Scale-invariance is a property also shared by fractals. When I first heard about Voss and Clarke’s study, I jokingly asked when we’ll be listening to fractal music. Needless to say, I learned that had been said and done when I was still wearing diapers. Google “fractal music” to see where this thought leads you.

I’m not sure what the power-law finding means for the origin of music, but intuitively what it means for what you hear is that music (at least the type we find appealing) lives on the edge between predictability and unpredictability. White noise has a constant spectral density and no autocorrelation. A random walk moving a melody along adjacent pitches has a strong correlation and a 1/f² spectrum. Somewhere in between are Bach and Adele.

When you turn on the radio, you want to be surprised – but not too much. Popular music today follows quite simple recipes. In most cases, you’ll be able to sing along when the chorus repeats. If you’ve heard a song a dozen times it gets dull though – it’s become too predictable. Symphonies are more complex, but they all have recurring motives and variations around that.

However, the musical edge must have a finite width. For some purposes, music can be more predictable than for others. What amount of predictability we find appealing doesn’t only depend on the occasion, it is also individually different. If you spend a lot of time analyzing pop songs, I suspect what’s in the charts today will sound very repetitive to you, though for the casual listener it arguably has an appeal.

It is tempting to extrapolate this to areas where autocorrelations are less easily measureable than pitch, for example to ideas in the written and spoken form. A scientific paper or a talk needs to strike a balance between the known and the unknown. Repeat too much common knowledge, and you’re obvious. Jump too far, and you’re crazy. The scientific pop stars are the ones on the edge. That also means the pop stars are the ones not too far ahead of their time.

It seems to me today the width of the scientific edge is very thin, maybe too thin. Sometimes, the obvious must be stated just so it remains in awareness. And sometimes the crazy starts making sense if you’ve listened to it often enough.

There’s another lesson. While fashions seem to come back, the cycle never is perfectly periodic, but always comes with a new twist. Thus, when the colors of the 70s will return to haunt us, maybe they’ll come with a metallic shine. And so, my impression that we’re having the same discussions over and over again must be wrong. They can’t be periodic, I am missing a change on longer time scales. History may be self-similar, but it’s not repeating. Though that's one of my all-time favorite songs.