Bach to chaos: chaotic variations on a classical theme - chaos theory used to compose music

Science News, Dec 24, 1994 by Ivars Peterson

Point by glowing point, the image swirls into view. As it builds up on the computer screen, it begins to resemble a delicate, stylized butterfly with translucent wings held lazily askew.

It's called the Lorenz attractor, named for meteorologist Edward N. Lorenz of the Massachusetts Institute of Technology, who in 1963 discovered this curious form encoded in a set of equations describing air flows in the atmosphere. The computer image arises out of a chaotic -- in the mathematical sense -- system.

For a given starting point, the computer calculates the coordinates of each successive point as the dynamical system described by the equations evolves. It displays these points as luminous dots on the screen. They appear to sprinkle themselves randomly across the display, but gradually a distinctive butterfly pattern emerges.

Different starting coordinates typically lead to radically different sequences of calculated points. But the overall pattern can always be identified as the Lorenz butterfly. It's an example of both the sensitive dependence on initial conditions and the distinctive patterns that are characteristic of chaotic systems.

When Diana S. Dabby, a graduate student in electrical engineering at MIT, first saw the Lorenz attractor a number of years ago, she was struck by its delicate beauty and elegance. "It appealed to my artistic side," Dabby says.

As a professional concert pianist and composer, she could envision "riding the back of the attractor" to create musical variations that stray in unexpected ways yet do not wander so far as to lose all ties with the original music. She could imagine using the mathematics of chaos to construct a musical space within which to work, create, and play.

As a first step toward realizing such an environment, Dabby has devised a scheme for using the Lorenz attractor to generate variations on the sequences of notes in a piece of music. She described her initial experiments -- done on Bach's Prelude in C from the first book of The Well-Tempered Clavier -- at an Acoustical Society of America meeting, held last June in Cambridge, Mass.

"My vision for this work is to expand it in every way to make a truly dynamic music for the future -- one that is always changing, but changing in musical, not random, ways," Dabby says.

Adecade ago, there was no chaos (mathematically speaking) in Dabby's life. Her musical career was in full swing. She performed in concerts in the New York City area and abroad, she composed music, and she practiced.

One day, while at the Lincoln Center library in New York City, Dabby came across an issue of a journal devoted to computer music. She noted that nearly all of the contributions to this particular issue came from mathematicians, computer scientists, or electrical engineers. "I wondered what would happen if a professional musician acquired their tools," Dabby recalls.

While still playing concerts, she taught herself algebra and returned to college to study calculus and take other courses required for entry into engineering. "I started to get ideas for new music," Dabby says. She also did well enough to get into the electrical engineering department at MIT, where she became a graduate student in 1987.

That fall, she encountered chaos in a course on dynamics. It caught her attention. "Chaos has a rich structure that is continually varying," Dabby says. "For a musician, this is much of the essence of music. There are themes, slight variations, and great variations."

"I started to learn everything I possibly could about chaos," she adds. "I went into the hardware [electronic circuitry]. I went into the theory. Anything that I could possibly do at MIT that had to do with chaos, I did."

In the spring of 1993, Dabby composed and performed a piece of music representing a sonic tour of Manhattan. It featured a piano and roughly 200 percussion, orchestral, and electronic sounds.

In her composition, Dabby sought to give the illusion that the piece was always changing with time. She created multiple paths through the music so that on successive hearings, a listener could always choose to follow different musical threads.

It is not unlike, but more complex than, following the flute's melodic line in the midst of a symphony.

"In this way, the piece appeared to vary from one hearing to the next," Dabby says. "But in fact, it did not do so. All of the electronic parts were fixed on tape, and the piano part was written out, not improvised."

The process of composing this music led Dabby to think about ways of creating variations -- changes in the sequences of notes -- such that a piece of music could actually differ from one hearing to the next. She ended up inventing a procedure based on the characteristics of a chaotic system to generate these different versions.

The x coordinates of the points that make up the Lorenz attractor for a given starting point fall within a certain range of numbers. Dabby's idea was to list the pitches of all the notes or chords of a musical piece and assign them one by one, in order, to the x coordinates of points belonging to the attractor. In this way, she paired up each of the pitches in the original music with a particular range of x values.