music from numbers

Algorithmic composition and musical sonification

     For the past 15 years, my electronic music has focused on creating music from numbers. I use computers to turn mathematical ideas–number sequences, equations, data, and so forth–into music. Each composition begins as an experiment in sound that asks the question:

What would this mathematical idea sound like?

All of the works on this CD are algorithmic compositions (Essl 2007) based on isomorphisms between numbers and sound. In the words of cognitive scientist and Pulitzer prize-winning author Douglas Hofstadter, an isomorphism is an “information-preserving transformation" (Hofstadter 1999).

The word ‘isomorphism’ applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where ‘corresponding’ means that the two parts play similar roles in the respective structures .

Any mathematical function may be turned into an algorithm (a step-by-step procedure that describes a task in a finite number of steps) capable of generating music. It is also important for the listener to know that a wide variety of real-time interaction models between composer, performer and machine (Jordà 2007) were employed in the realization of these works. Some of these approaches are described in the notes that follow.

     The album’s title, Sounding Number, was derived from numerus sonorous, a Latin term used by Renaissance music theorist Gioseffo Zarlino to refer to the neo-Pythagorean belief in an almost magical relationship between music and number (McCreless 1997). The emerging science of auditory display was also a source of inspiration and ideas. Just as one graphs an equation like y=x2 in the geometric plane, one may alternatively sonify the equation in the sonic domain in order to learn more about its structure. In many cases, the ear can perceive gradations not perceivable by the eye. In the field of auditory display, the term sonification is used to refer to the process of mapping data to sound for scientific purposes (Worrall 2009). I use sonifications, however, as the basis for musical compositions (Bain 2006).

     The first step in my compositional process is to use a musical programming language (Max/MSP, Csound, or SuperCollider) to write a computer program that allows me to efficiently explore the myriad ways a particular mathematical idea may be turned into music. Dozens of trials and corresponding revisions to the program are usually required before the experiment begins to produce interesting musical results. Numbers are strategically mapped onto musical parameters such as pitch, intensity, timbre, and spatial location in order to create musical ideas. Once the initial experiment yields what I deem to be viable musical results, I tear down the experimental scaffold and build a new, more complex, program that allows me to interactively collaborate with the computer to compose a full-length composition based on these musico-mathematical object trouvés. I call this compositional process musical sonification. As you listen to the album, you will encounter musical sonifications of prime numbers, Fibonacci numbers, triangular numbers, the golden ratio, the logarithmic spiral, fractals, chaos theory, the digits of pi, and more. I hope you enjoy the works on this CD and the mathematical ideas that inspired them, ideas encoded within the many levels of the music’s structure.


Bain, Reginald, 2006. “The AIMS Project: Creative Experiments in Musial Sonification,”
     in Proceedings of the 2006 International Computer Music Conference, New Orleans, LA.
Essl, Karlheinz, 2007. “Algorithmic Composition,” in Collins, Nick and Julio d’Escriván, eds.,
     The Cambridge Companion to Electronic Music, pp. 107-125.
Hofstadter, Douglas, 1999. Gödel, Escher, Bach: An Eternal Golden Braid, 20th Anniversary ed.
      New York: Basic Books, 1999), pp. 9 & 49.
Jordà, Sergi, 2007. “Interactivity and Live Computer Music,” in Collins, Nick and Julio d’Escriván, eds.,      The Cambridge Companion to Electronic Music, pp. 89-106.
Kramer, et al., 1994. Auditory Display: Sonification, Audification, and Auditory Interfaces:
      A Proceedings Volume in the Santa Fe Institute Studies in the Sciences of Complexity,
      Proc. Vol. XVIII. Reading, MA: Addison-Wesley.
McCreless, Patrick, 1997. “Rethinking Contemporary Music Theory,”
      in Keeping Score: Music, Disciplinarity, Culture. Charlottesville, VA: University of Virginia Press, p. 24.
Worrall, David, 2009. “An Introduction to Data Sonification,” in Dean, Roger T.,
     The Oxford Handbook of Computer Music. New York: Oxford, pp. 312-33.


Cycling ’74's Max/MSP.
Kramer, G., et al. 1997. Sonification Report: Status of the Field and Research Agenda
Polansky, Larry, 2002. Manifestation and Sonification: The Science and Art of Sonification:
      Tufte’s Visualization, and the “slippery slope” to Algorithmic Composition (An Informal Response
      to Ed Childs’ Short Paper on Tufte and Sonification).
Vercoe, Barry, et al., Csound.

Updated: January 12, 2012