Musical numerology #2 ~ Fractal Music and Algorithmic Composition

The Claim: algorithms like fractals, L-systems and statistical models can be used as musical tools on many levels. Statistical analysis reveals the inherent fractal nature of music (1/f noise). Fully automated, computer-generated algorithmic compositions, created with minimal human intervention, qualify as real music.

The evidence: musical arguments

Complexity is the great new scientific paradigm of the 21st century. The mathematical branch of this new approach to science was first noticed by the general public when striking computer-generated images of fractals and chaotic behaviour were published during the early 1980s. These images suggested that God is a hippy after all.

Fractal sets are generated using simple algorithms; they are dynamic systems powered by feedback loops. The colorful computer images produced by fractal sets look deep because of self-similarities in the structures, and they look complex because of a certain amount of variation created by the feedback loop. Unlike oldskool Euclidean geometry, fractals seem to capture the richness and variety of forms found in nature. But just because a computer image of a fractal set looks good (for a graph) that doesn't mean that it will sound good.

The guy in the video, Harlan Brothers, is a mathematician. He's into "a rigorous mathematical approach" of fractal music, which sounds appropriate, although you could accuse Mr. Brothers of having a mechanical, or even somewhat totalitarian view of creativity. (Brothers isn't really a mad scientist but he does own a few funny patents like the smokemate, the opticord and the bathtub buddy.)

Mathematicians like Brothers love to talk about Bach. There has been a substantial Bach number-cult for centuries, and in recent times fractals have been added to the repertoire. Indeed, there is a suggestive kind of similarity between the parts-resembling-the-whole-character of fractal graphs and the theme-and-variations-structure of certain musical styles, like baroque music. But it's not really the same thing. Sorry Mr. Brothers, just trying to be "mathematically rigorous". Sierpinski triangles sound especially boring..

On the other hand, an experienced composer like Bach, who knows about themes and variations and musical structure, is able to use virtually any kind of theme and turn it into music. So why not use the number pi or e, or fractals and algorithms as themes? No problem at all for someone like Bach.

The only real problem for fractal-based music or formal algorithm-based music is this: how do you map the raw numbers of - say - a Mandelbrot-graph to corresponding musical events (time, pitch, loudness, timbre etc.)? There's no consensus about the best approach to mapping. That's good, because if you want, you can be really creative and come up with your own mapping styles.

OK, now let's say that you're not an experienced music composer. You are not Bach, you are a mathematician or a biologist interested in sounds and numbers and sonification. How do you give your number mapping-project some musical value, considering you don't have compositional skills?

First of all, you're working with plain and simple numbers, and while these curly little symbols are useful for calculating and so on, they don't have intrinsic sonic or musical qualities. What does 5 or 1/666 or pi or 43645 sound like? Hard to say unless you're a synesthete.

The straightforward mapping of numbers to 'midi note numbers' (where you map a number to the standard pitch of an existing sound, like a digital piano patch) is the most common (and boring) type of mapping. You end up with sonic statistics. Musical mapping turns out to be more difficult than it would seem. Especially if you want more than semi-random bleeps and noise. From a data set point of view the generation of 'real music' is non-trivial to say the least.

While the sonification of a given set of numbers can be problematic if you want the results to sound good, at least you know right from the start what set of numbers to use. This means you limit yourself in a way that can produce creative solutions, if you try hard enough.

On the other hand you have algorithmic composers who don't start with a given set of numbers. They just assume that there are data sets 'out there' that (should be able to) sound good. A common strategy used by these folks is the use of 'vaguely interesting' numbers: successful biological algorithms, weather statistics, esoteric interplanetary ratios etc. This is confusing: what's the point of a musical translation of a DNA sequence? Why should a random piece of DNA sound better than random phone numbers?

There's yet another catch for fractal music composers: the recursive quality of fractal sets implies that they could, or maybe even (if you're trying to be rigorous) should show up on any 'scale' or level of the fractal-musical process/experience. The mapping problem again: on what level or levels can fractal sets function in music? Are we talking composition? Sound design? Performance variables? What does a rigorous implementation of fractal sets in music imply about the mapping of a given set on the various levels of sound and music?

Anyway, here's a suggestion for fractal music composers that's usually ignored: why not look at dub music? The recursive quality of fractal sets is also found in the feedback loop of sound effects used in dub. Most effects (echo, reverb, phasing, flanging, distortion etc.) have a feedback button. But keep in mind that soundcard latency and feedback don't mix. That's why for sound effects, analog techniques are often preferred to the more sterile feedback that's calculated by computers.

The evidence: numerological arguments
The famous sceptic, magician and mathematician Martin Gardner has said that most music we listen to is 1/f noise. What he means by this, is that most music has the right mix of patterned predictability and random unexpectedness, which is pleasing (as in: not boring) to the human ear.

And as a kind of bonus, if you look at the shape of the curve described by this so-called 1/f music, it has a fractal shape. That's weird! This means that humans have been creating fractal music for centuries without even knowing it (no computers needed). If you know where to look, fractals are as common in music as the Fibonacci series. Scientists love this. Fractals have been explored in music and sound since the 1980s. Recently fractal structures in melodies (Mason and Saffle, 1994; Chesnut, 1996) and in musical forms and phrase structures (Solomon, 1998) have been researched. Among the fractals identified in musical structures are Sierpinski triangles, Peano curves and Koch snowflakes.

But keep in mind that most scientific research into the fractal properties of music is statistical, descriptive and after the fact. In other words: it doesn't help a mathematically rigorous algorithmic music composer, unless the explicit goal is to create statistically correct music (sounds like airports and elevators).

The verdict
An experienced musical composer like Johann Sebastian Bach can use any kind of raw material and turn it into music. This includes algorithms, fractals and all kinds of 'vaguely interesting numbers'. Having said that, algorithms, fractals and numbers have no inherent musical qualities. Even if the approach is mathematically rigorous, fully automated (algorithmic) music composition can only hope to be statistically correct, not aesthetically correct. The recursive nature of the fractal algorithm is no replacement for themes and variations.

1. Visit an aquarium and observe very closely. Compose a piece of music based on your observations and impressions.
2. Write an algorithm that mimics the synchronized movements of a school of fish. Map the movements of the fish to the feedback buttons of your sound effects.
3. Watch this Benoit Mandelbrot video and write a really rough tune.

4. The philosopher Richard Rorty saw the idea of mathematics as a 'language of nature' as pervasive throughout the history of western philosophy. Using the 'language of objectivist metaphysics', explain how this inevitably leads to attempts at making algorithmic computer music.