The theory of quantum mechanics introduced an unavoidable element of unpredictability or randomness into science. In 1922, Albert Einstein was awarded the Noble Prize in physics for his contribution to quantum theory, yet Einstein never accepted the notion that the universe was governed by chance. His feelings were summed up in one of his most famous statements: “God does not play dice!”

     Echoing the spirit of non-deterministic behavior characteristic of the sub-atomic world, all of the music is created using chance procedures. Each movement is a compositional experiment in which various types of scaling noises are used to create random melodies. The simplest example of a scaling noise, white noise, is most commonly experienced as the thermal noise produced by a radio when there is no input signal. Einstein himself contributed to our understanding of Brownian motion, a type of noise associated with the random motions of small particles suspended in a liquid. More recently, scientists have discovered one-over-f noise (1/f noise), self-similar random fluctuations that have reportedly been found in the annual flood levels of the Nile river, the U.S. stock market, the flow patterns of traffic on an expressway (Mandelbrot 1977), and in music (Voss and Clarke 1978). White, brown, and 1/f noise may all be simulated on a computer using dice-rolling algorithms (Gardner 1992).

      When studying a fluctuating quantity like noise, scientists often analyze its spectral density. White noise is said to have a spectral density of 1/f⁰ because it produces an uncorrelated sequence of values. Brown noise on the other hand, which has a spectral density of 1/f², is quite correlated. 1/f noise is often viewed as a compromise midway between white and brown noise (Gardner 1992). 1/f distributions were used to determine all of the pitches in this work. For example, in the first movement the output of the 4-dice version of R.F. Voss’ 1/f algorithm was mapped to a C-major collection in order to create 16-note melodies (Bolognesi 1983). After numerous experimental trial runs of the algorithm, I finally discovered the melody shown in Fig. 1–which was to my liking.

Fig. 1. The first movement's 1/f theme.

Note the use of fixed quarter-note durations. It should also be mentioned that brown-noise distributions were used to sketch in other parametric details of the theme. Fig. 2 shows the Max/MSP code that calculated the pitches of the melody in Fig. 1.

Fig. 2. 1/f algorithm implemented in Cycling 74's Max/MSP.

      Each movement takes its title from an important idea of quantum physics: the first movement from Werner Heisenberg’s “uncertainty principle” (we cannot know the exact position and velocity of a particle at the same time); the second movement from the paradoxical thought experiment commonly referred to as "Schrödinger’s cat" (from the point of view of quantum mechanics, the cat in the experiment may be simultaneously viewed both dead and alive); and the third movement by Richard Feynman’s "sum over histories" (a particle travels along every possible path between points A and B). The three movements are arranged in a fast-slow-fast arch structure that may be described as: I. 1/f theme and variations, II. 1/f chaconne, and III. 1/f canon.

RB

references

Bolognesi, Tommaso, 1983. "Automatic Composition: Experiments with Self-Similar Music,"
      Computer Music Journal, Vol. 7, No. 1, pp. 25-36.
Gardner, Martin, 1992. "White, Brown, and Fractal Music," in Fractal Music, Hypercards, and More.
      New York: W.H. Freeman.
Hawking, Steven, 2001. The Universe in a Nutshell. New York: Bantam, pp. 83-85, 147-48.
Mandelbrot, Benoit B., 1977. The Fractal Geometry of Nature. New York: W.H. Freeman.
Penrose, Roger, 1989. The Emperor's New Mind: Concerning Computers, Minds,
      and The Laws of Physics. New York: Penguin, pp. 248-50, 290-93.
Voss R.F. and J. Clarke, 1978. "'1/f Noise' in Music: Music from 1/f Noise,"
      Journal of the Acoustical Society of America, Vol. 63, pp. 258-263.

Updated: January 12, 2012