Fractal Dimension of Music

Is music fractal? Yes!

Fractal dimension is a way of talking about things that seem to break the normal rules of geometry. The dimension of music audio is between 1 and 2: between a flat line and a two dimensional space. According to this paper, it is more specifically between 1.7 and 1.9. Different genres have different values, with rock having a higher value than classical, for instance.

Paper: “Fractal dimension and classification of music”, Bigarelle and Alain, 2000, available here.

Impossible music

Benedetti’s puzzle (mathematically impossible music) by Adam Neely

A good example of how our familiar 12 tone pitch system is an elaborate mathematical compromise.

In my fractal music I use a system of multiplying the pitch parameters by whole numbers. So one note might be 2*2, and another 2*3, another 3*3. If an oscillator is tuned to these ratios directly, they are in strict mathematical relation, not using the adjustments for equal temperament. For instance this is built of a layering of many pure sine waves, which build up an effect of timbre through many overtones.

Music at the Edge of Chaos

Fracas by Shawn Bell, 2020

NetWorks is a music-generating system and a body of work based on Network theory, Complex Systems Science and the Honing Theory of creativity (Gabora 2010). NetWorks uses a hierarchical, scale-free network to generate music that can range from orderly to chaotic. At the ‘edge of chaos’ it generates patterns that exhibit emergent complexity through coherent musical development at low, mid and high levels of musical organization, and often suggests a sense of direction through goal seeking behaviours. …

In general, the sounds chosen to manifest the musical patterns discovered by a network attempt to reflect the mystery and wonder that virtually unlimited diversity can come from such simple models of complexity. When mapping patterns to sound, every effort is made to preserve the integrity of the patterns rather than obfuscate them with complex synthetic textures or other effects that are readily available during mixing.

Shawn Bell

Darwin Tunes

To investigate the role of consumer selection, we constructed a Darwinian music engine consisting of a population of short audio loops that sexually reproduce and mutate. This population evolved for 2,513 generations under the selective influence of 6,931 consumers who rated the loops’ aesthetic qualities. We found that the loops quickly evolved into music attributable, in part, to the evolution of aesthetically pleasing chords and rhythms. Later, however, evolution slowed. 

MacCallum et al, Evolution of music by public choice.