Music From Numbers

Album Cover | Tray Card

     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 based on isomorphisms between numbers and sound. Any mathematical function may be turned into an algorithm 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 were employed in the realization of these works. Some of these techniques are described in the notes that follow.

     The album’s title, Sounding Number, was derived from the Latin term numerus sonorous. Renaissance music theorist Gioseffo Zarlino used it to refer to the neo-Pythagorean belief in an almost magical relationship between music and number. The emerging science of auditory display was also a source of inspiration and ideas. Just as one graphs an equation in the geometric plane, one may sonify an equation in the domain of sound 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. I, however, use sonifications as the basis for musical compositions.

     The first step in my compositional process is to use a musical programming language (I used Max/MSP and Csound on this album) 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 required before the experiment begins to produce interesting 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, algorithmic implementation that allows me to interactively collaborate with the computer to compose a full-length composition. I call this computer-aided 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.


the music of the primes

     The Music of the Primes is a musical exploration of the spacing of prime numbers, numbers that cannot be expressed as the product of two smaller numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …

In his 2003 book of the same title, mathematician Marcus du Sautoy describes the primes as the “atoms of arithmetic,” the “mathematician’s periodic table,” the “irregular heartbeat of mathematics,” and even more prosaically, as “jewels studded in the infinite universe.” From its use of a prime-inspired just intonation to its use of repeated rhythmic cycles based on primes, the pitch and rhythmic domains of this work are saturated with various musical manifestations of prime spacing. The Greek mathematician Euclid proved that there are infinitely many primes. In many ways this work is an attempt to make the infinite finite and musically engaging. It is an homage to Euclid, Eratosthenes, Gauss, Riemann, and other mathematicians who have contributed to our appreciation of the beauty, mystery, and music of the primes. (top)

the butterfly effect

     In his 1987 book Chaos: Making a New Science, author James Gleick describes The Butterfly Effect as “…the notion that a butterfly stirring the air today in Peking can transform storm systems next month in New York.” This work attempts to breathe life into that metaphorical butterfly. Gleick’s quotation captures the spirit manifest in the theory of sensitive dependence on initial conditions first described by scientist Edward Lorenz. Lorenz was using primitive computer models to investigate his intuitive notions about the order he perceived in seemingly disorderly weather systems, when he discovered that a small adjustment in the input of a system could produce large-scale consequences in the output. Represented in the work by a continuously sweeping glissando gesture, the butterfly’s improvised flight is controlled by a live performer who interacts with the Doppler equation to control the pitch trajectory of the glissando.

The butterfly is set against a deceptively simple sonic landscape, a Fibonacci-inspired ostinato pattern whose spectral content and spatial location are in constant flux. (top)

chaos game (for nancarrow)

     Chaos Game (For Nancarrow) is a musical implementation of the triangular version of Barnsley’s well-known chaos game algorithm. Mathematician Michael Barnsley coined the term to refer to a process by which a fractal image is generated on a computer screen using randomly chosen points. This work is a musical fractal whose fabric is woven by mapping points on the computer screen to notes that are heard in real time. Because the pitch selections are essentially random, every performance of the work is quite different at the musical surface. At higher levels of structure, however, scale sieves and other boundary conditions constrain the random selections in ways that provide the listener with a familiar sense of harmonic direction across multiple performances. The surface detail of this fractal-music tapestry is created by two-voice deceleration canons, a Nancarrow-inspired micropolyphonic technique that features a shifting time interval between two canonic voices. This shifting time relationship creates a sense of curved time between the voices, a musical analogue of a tessellation process that, at times, produces paradoxical resultant patterns. Finally, in keeping with the triangular theme, all of the rhythmic parameters were derived from triangular numbers. (top)

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …

"god does not play dice!"

     The theory of quantum mechanics introduced an unavoidable element of unpredictability or randomness into science. In 1921, Albert Einstein was awarded the Nobel 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, and in music. White, brown, and 1/f noise may all be simulated on a computer using dice-rolling algorithms. 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 as being 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. (top)

when inspiration came

     Experimental music composers and scientists share a similar process of discovery. When Inspiration Came also takes its name from a line in James Gleick’s book Chaos: Making a New Science. Gleick provides a compelling account of how scientists like Mitchell Feigenbaum and Robert Stetson Shaw used computers in the 1970s to explore the boundary between order and chaos. Feigenbaum’s study of the logistic equation (a simple chaotic model for population growth) and Shaw’s experiments with a dripping faucet led to important discoveries in chaos theory. I have always been fascinated by the sound of a dripping faucet. Its steady beat with unanticipated syncopations can be mesmerizing. When the drips fall into a small pool of water this natural process occasionally produces a very interesting melody. This composition is a sonic journey through the logistic equation. 

The pitches of the drips are determined by te equation, and granular synthesis techniques slowly turn a single steady drip into a “rainstorm” and back again. The granular microsurface mirrors the bifurcations and labyrinthine complexities of the logistic equation. Beneath this surface, a supporting chorus of four pink-noise ostinati sing randomly-selected harmonic partials. Contrastingly, the work’s macrostructure is determined using the golden proportion.

the language of the angels

     The Language of the Angels was inspired by Olivier Messiaen’s invention of a communicable language. The privileged wordless communication of angelic speech described by Saint Thomas Aquinas in his Summa Theologica (c1265-1274) was Messiaen’s inspiration for the playful invention of a musical alphabet for his solo organ cycle Méditations sur le Mystère de la Sainte Trinité (1969). This alphabet of sounds, in which each letter of the alphabet is mapped to a specific pitch and duration, allowed Messiaen to transcribe words into music. The language serves as the core element in his vivid musical depictions of angelic communication. Aquinas describes the speech of the angels in the five articles of question 107. The first movement of the work is a musical depiction of the first article. Based on the second article, the second movement introduces the main theme of the work–an ostinato figure that, like the material in the first movement, is subjected to algorithmically-generated antiphonal exchanges.

Whereas the first two movements feature the performer playing in a traditional manner on a synthesizer keyboard, the third movement is performed on a computer keyboard. An algorithmic implementation similar to Messiaen’s communicable language allows the performer to type words on the computer keyboard and have these improvisations immediately turned into music. The fourth and final movement is a depiction of the fourth and fifth articles. It recapitulates and combines musical elements from the three previous movements. (top)

strange attractors and logarithmic spirals

     Conceived for the digital audio signal processing environment Csound, Strange Attractors and Logarithmic Spirals sets into opposition sonic manifestations of two beautiful mathematical forms: strange attractors, chaotic systems that cycle periodically yet never repeat exactly the same pattern; and logarithmic spirals, a familiar shape found throughout nature in shells, tusks, sunflowers, galaxies, and so on. The work features two contrasting approaches to synthesis: one inspired by an instrument design by Csound guru Hans Mikelson, the other by music-scientist Jean Claude Risset. The Mikelson instrument is a sonification of the Lorenz attractor, a chaotic system of three differential equations discovered by scientist Edward Lorenz.

When initialized with a special set of values, the values used in this composition, a graph of the system produces a beautiful butterfly-shaped pattern. The sound quality of the sonified Lorenz attractor varies from noise-like pulses to percussive zips and buzzes. The Risset instrument, on the other hand, generates sounds of an entirely different nature. Risset’s ingenious additive synthesis technique produces a cascading arpeggio in the harmonic series, a technique that offers the composer precise control over the arpeggio’s pitch content and rate of speed. The sustained timbres produced by this instrument range from organ-like sounds to the sounds produced by a Tuvan throat singer. All of the musical parameters of this composition are derived from the Fibonacci sequence.

1, 1, 2, 3, 5, 8, 13, 21, …

Its connections to the logarithmic spiral and to the golden proportion are also explored. This so-called divine proportion, which has a distinguished history as a source of inspiration for artists, is expressed in the music in myriad ways including as the pitch interval (1.618:1) spanned by many of the work's glissandi and as the key to which the work modulates. (top)

pi day

     On March 14 (written 3/14 in America) math enthusiasts all over the world celebrate Pi Day, a holiday dedicated to the world’s most famous mathematical constant. Denoted by the Greek letter π, this ratio is ubiquitous throughout science and mathematics. For any circle, if you divide its circumference by its diameter you will always get the same decimal value, a sequence of digits that never repeats and has many interesting properties.


The composition is my answer to the call for silly salutes to this mysterious number. A computer-generated blues in Bb major, the music is algorithmically generated by a computer program that maps the digits of pi to pitches in real time. Following a brief slow introduction, all of the music is sculpted from the first 512 digits of pi. The digits are strategically mapped to pitches, and other musical parameters, at a constant tempo and pulse duration in order to create interesting musical lines. The lines are then layered using an additive formal process: first a log drum enters with a steady eighth-note pulse, then a bass line with an idiomatic groove, then woodblocks in canon, then a 12-string guitar takes over, … Contrastingly, the sampled voice part (spoken by my wife, Erin Keefe Bain) recites the digits at a slower pi-related tempo. The irrational polytempo relation between the foreground music and vocal recitation creates a subliminal rhythmic tension that is not resolved until the voice part begins its infinity implying fade. (top)

Updated: January 12, 2012