Fractal Music

Pulse Trains

Depending on the genre, the arrangement, the instrumentation, and the performance, many different frequency pulses can propagate through a single piece of music.
We will refer to these propagating waves of rhythm as pulse trains.

In any given song, it is the various compositional elements (i.e., note values, chordal changes, lyrics) combined with the interplay of the instruments that gives rise to the perceptual appearance of pulse trains.

Pulse trains inhabit our perceptual space.

Consistent with the propagation of sound waves, longer pulses can be thought of as generally possessing more power and carrying a longer distance before fading away. We will refer to this aspect of perceived power as depth. Greater depth is equivalent to longer persistence.

Consider the short-lived effect of a quick click of a castanet as compared to the more enduring effect of a booming contrabass whole note. A lower frequency pulse train associated with the bass voice will have greater depth than the higher frequency pulse train associated with the castanet.

Common time (4/4) lends itself well to multiple pulse trains whose relative frequencies are powers of 2.

Combining the idea of multiple, simultaneous pulse trains with the idea of persistence, we can draw a graph representing a collection of pulse trains in 4/4 time. Here is an illustration

This graph represents six pulse trains, each consisting of a series of pulses of equal depth. Pulses that coincide with larger pulses are hidden. Beginning with the highest frequency, each pulse train is twice the depth and half the frequency of the one preceding it.

The most significant characteristic of this graph is that, over its limited range, it exhibits scale invariance. Furthermore, a log-log plot of pulse count against either frequency or depth demonstrates the existence of a power law relation.

These pulse trains are fractal.

Return to Pulse Trains.

© 2004 Harlan Brothers