From Xenakis to Wolfram: Cellular Automata and its Application to Music

Katarina Miljkovic (New England Conservatory of Music, USA)

“Another approach to the mystery of sounds is the use of cellular automata which I have employed in several instrumental compositions these past few years. This can be explained by an observation which I made: scales of pitch (sieves) automatically establish a kind of global musical style, a sort of macroscopic ‘synthesis’ of musical works, much like a ‘spectrum of frequencies, or iterations’ of the physics of particles. Internal symmetries or their dissymmetries are the reason behind this. Therefore, through a discerning logico-aesthetic choice of ‘non-octave’ scales, we can obtain very rich simultaneities (chords) or linear successions which revive and generalize tonal, modal or serial aspects.” (Xenakis, Formalized Music, xii)

The preceding quote asserts Xenakis’ interest in the application of Cellular Automata (CA) to music and his vision of its impact on musical language. His observation about an automatically established “global musical style” that would generalize “tonal, modal or serial aspects“ through CA-generated frequency spectra, is intriguing, far reaching and very close to Stephen Wolfram’s work.  In his New Kind of Science, Wolfram establishes elementary rules for CA as a global language for communicating and exploring a "computational universe of simple programs that can capture the essence of the complexity--and beauty--of many systems in nature.”

Wolfram’s fascinating palette of elementary rules, computer graphics and sound textures, generated in his software Mathematica, recalls the following statement by Xenakis: “Today, there is a whole new field of investigation called Experimental Mathematics, that gives fascinating insights into automatic dynamic systems by the use of math and computer graphics. These studies lead one right into the frontiers of determinism and indeterminism” (Formalized Music, xiii).   Wolfram’s insistence on three modes of perception of CA: conceptual, visual, aural and, an easy transfer from one to another, is the integration we find at the core of Xenakis’ work and his legacy.  Further, Wolfram’s classification of 256 elementary rules according to degree of complexity into four classes (simple, periodic, random and universally complex) stimulates experimentation with the transition from “chaos to symmetry and the reverse orientation” (ibid., xiii), in a new and insightful way.  

As a composer, I have always been fascinated and inspired by the work of Iannis Xenakis, so it seemed inevitable that I would come across Wolfram’s research.  About a year ago, I took up an exploration of New Kind of Science and, together with members of Wolfram Research team, extending the application of Elementary Rules to music.  This presentation is an attempt to collect and summarize my ongoing experiences in sound experimentation within the framework of New Kind of Science.  I’ll concentrate on three main points.  First, through the mapping of representative rules into a pitch scale, vertical sonorities, rhythm and sound color, I’ll demonstrate that each of the four classes generates its own musical style through a characteristic rhythmic activity and frequency spectrum.  I will explain the mapping decisions, which has proven to be a nontrivial issue.  Second, I’ll examine variations of music material produced by the elementary rules within the same class and point to similarities and differences in their spatial and temporal aspects.  Third, I’ll demonstrate that an endless number of permutations can be generated from a single rule by changing its initial conditions.  At the end of my presentation, I’ll address the implications of the preceding points for further research, composition, and performance in real time in Mathematica.

Composer Katarina Miljkovic was born in Belgrade, Serbia (1959).  In 1992 she moved to Boston, MA where she has been teaching at New England Conservatory of Music since 1996.  Miljkovic has written for symphony orchestra, string orchestra and various other groupings, including works for saxophone, electric guitar and computer sounds.  Ms. Miljkovic has been exploring the relationship of nature and music composition.  This initially led her to the Benoit Mandelbrot’s essay The Fractal Geometry of Nature, and self-similar systems.  The study resulted in her cycle, Forest, for two prepared pianos and percussion, released by Sachimay Records.  Currently, Ms. Miljkovic is working on mapping Elementary Rules of New Kind of Science, by Stephen Wolfram, to sound.  She presented her exploration in this new field at the International Conference, NKS 2004, Waltham, MA and NKS Summer School 2004 at Brown University, Providence, RI.