The word “chaos” is often used in common parlance to mean things that are disordered. To mathematicians, chaos is behavior that appears random, but is actually the result of a system that is evolving according to set rules. The reason chaotic systems seem so unpredictable and random is that they are sensitive to slight changes in initial conditions, commonly referred to as the butterfly effect—the idea that something as small and unpredictable as a whisper of air movement from an insect flapping its wings could cause a thunderstorm halfway across the globe.
Dabby became interested in something called the “strange attractor,” a mathematical entity that pulls trajectories back toward it, sort of like a whirlpool that draws in nearby currents. Those trajectories are similar, but differ depending on the initial conditions. It reminded her of the musical concept of theme and variation—think of the recurring motifs in jazz, or any number of classical pieces where the theme is played, then repeated and translated into something new. What if, Dabby wondered, it were possible to generate chaotic variations in music?
Strogatz admits to being a little skeptical of his unusual student at first. She was, he recalls, eager and engaged, but weak by any conventional measure.
“She was really noticeably different from the other MIT students, in that she could barely do algebra; she was really mathematically in a different place than everybody else,” Strogatz, now a professor at Cornell University, recalled. “I knew she was enthusiastic, but the jury was still out on whether she could produce anything good.”
For her graduate work, Dabby used chaotic trajectories as a kind of scaffold for music. She would generate a particular curve, then translate a melody onto the curve—meaning that as one moved along the curve, each spot would correspond to one note. She would then map the notes onto a “pitch axis,” a kind of musical yardstick divided not into inches or centimeters, but into notes—one spot is middle C, another A sharp. Then, to create variations, she would generate a new curve, and use the pitch axis to determine the new order of notes. The result would be a subtly changed piece of music, with little and big differences in each phrase.
From the beginning, the technique generated music audio files—some of which Dabby learned to play on the piano. She wanted to show that it would work on all styles of music, so she applied it to a prelude by Bach and another by Gershwin. When she began to think about building a Web program open to anyone, she wanted to make it as open as possible—simply using music files that anyone might have on a computer to generate new music.
As time has progressed, so have the variation techniques, which don’t just rearrange existing notes, but play with rhythm and add new notes. The theory behind each technique is largely hidden, with effects described in simple language on the website. “Brought together by fate” alters the order of notes. “Twirling left” and “twirling right” generate variations that include notes not found in the original. “Dance all night” varies rhythms and pitches, building off the original and the variation as it unfurls.
Despite what you might expect, the results don’t sound like some new form of computerized robot music. Instead, the variations end up sounding familiar—each one resembling the original piece, but stippled with unexpected bits.
Strogatz said that despite his early doubts, Dabby’s work has been a legitimate contribution. The melding of art and math, he said, doesn’t always so successfully produce something that people in both fields can agree is interesting.
“I can’t tell you how many people send you files and say, ‘This one is a Lorenz attractor played through a loudspeaker,’” Strogatz said. “They’ll play chaos in the most literal way, and it’s bad. It’s uninteresting. Her work is musical. It retains and in some ways improves the musicality, but uses chaos in this interesting way.”
If chaos can be harnessed to generate new ideas in music, what other areas could be ripe for such perturbation? Liz Bradley, a professor at the University of Colorado Boulder who met Dabby when they were graduate students at MIT, talks about the work in a class she teaches on nonlinear dynamics, as a way to wake students up to the practical and creative applications of chaos theory.
“This is something that just hooks them immediately,” Bradley said. “It’s a fabulous way to make mathematics and computer science real and tangible, and exciting....You have a whole class of students and they’re like differential equations, big whoop, and then you show them this and they sit up.”Continued...