|Ivars Peterson's MathTrek|
January 11, 1999
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. The overall pattern, however, 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.
In the early 1990s when Diana S. Dabby was a graduate student in electrical engineering at MIT, she could imagine using the mathematics of chaos to compose music. She envisioned "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.
A concert pianist before she came to MIT, Dabby devised a scheme for using the Lorenz attractor to generate variations on the sequences of notes in a piece of music. Her initial experiments were done on Bach's Prelude in C from the first book of The Well-Tempered Clavier.
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.
Then she could choose a second starting point only slightly different from the first to produce a new "trajectory", or sequence of points, making up the Lorenz attractor. Because this new trajectory generally does not track the original one perfectly, the x coordinates of the two trajectories differ and the musical notes corresponding to these new x coordinates may occasionally change from those in the original piece.
One can imagine that the initial "mapping" step lays down the musical landscape, and the second trajectory, representing the variation, takes a slightly different path through this terrain. Because the landscape incorporates features of the original musical piece, any variations created in this way usually sound consistent with the source piece. Indeed, no variation can include a pitch not present in the original.
"The musical variations that result can be close to the original, mutate almost beyond recognition, or achieve degrees of variability between these extremes," Dabby notes. "The technique can also be used to infuse a given work with the attributes of another, so that, for instance, a work by Bach can acquire attributes of a work by Gershwin."
To Dabby, this method of producing musical variations serves as an idea generator. It brings a fresh perspective to familiar music. A musician can interact with the variation-producing software to select, edit, and record particularly pleasing passages, even weave them into new compositions.
Dabby has applied the technique not only to works by Bach but also to a Gershwin piece and some of her own compositions. Recently, she has explored the use of chaotic mappings to create rhythmic variations via the y variable of a Lorenz attractor.
"Once variations of an entire piece are available, the composition can change with successive listenings, from performance to performance, or even within the same concert," Dabby remarks. "In a broad sense, the music has become dynamic. It changes with time much the same way as a river changes from day to day, season to season, yet is still recognized in its essence."
Inspired by Dabby's example, computer scientist Elizabeth Bradley of the University of Colorado and Joshua Stuart, now at Stanford University, have developed a similar scheme for dance. They use chaos to generate variations on movement sequences.
"We map a progression of symbols representing the body positions in a dance piece, martial arts form, or other motion sequence onto a chaotic attractor, establishing a symbolic dynamics that links the movement progression and the attractor geometry," Bradley and Stuart report in the December, 1998 Chaos.
The researchers used special symbols to represent human body postures, encoding those positions by defining an axis and angle of rotation (given in the form of a mathematical expression called a quaternion) for each of the main joints. They then mapped a given motion sequence--whether a ballet jump or sequence of karate moves--onto a chaotic attractor. Following a new trajectory around the attractor produced a variation of the original motion sequence.
In effect, the choreographic software took an animation as input and generated an animation as output.
The researchers also had to adjust for the capabilities of the human body. "While musical instruments can play arbitrary pitch sequences, subject to instrument range and performer ability, both kinesiology and aesthetics impose a variety of constraints on consecutive body postures in dance and martial arts genres," Bradley and Stuart note. To smooth out abrupt transitions introduced by the chaotic mapping, the researchers developed schemes that captured and enforced particular dance styles.
Bradley's Web site at http://www.cs.colorado.edu/~lizb/chaotic-dance.html introduces their Chaographer software.
The researchers have shown the resulting animations to hundreds of people, including dancers and martial artists. "The consensus is that the variations not only resemble the original pieces but also are, in some sense, pleasing to the eye," Bradley and Stuart conclude. "They are both different from the originals and faithful to the dynamics of the genre. There are no jarring transitions or out-of-character moves."
They add, "Showing these results in a classroom is an enormously effective way to motivate students to learn the mathematics of rigid-body dynamics, chaos, and context-dependent grammars."
The Chaographer software might also have come in handy when my 11-year-old son Eric had to invent his own karate (tae-kwan-do) combination as part of his black-belt exam last summer. On the other hand, it wouldn't have helped me. I had enough trouble learning the required traditional and modern forms up to the red-belt level, which is as far I managed to get.
Copyright 1999 by Ivars Peterson
Bradley, E., and J. Stuart. 1998. Using chaos to generate variations on movement sequences. Chaos 8(December):800.
Dabby, D.S. 1996. Musical variations from a chaotic mapping. Chaos 6(June):95.
Lorenz, E.N. 1993. The Essence of Chaos. Seattle: University of Washington Press.
Peterson, I. 1994. Back to chaos. Science News 146(Dec. 24&31):428.
Additional information about the Chaographer software for generating choreographic variations is available at http://www.cs.colorado.edu/~lizb/chaotic-dance.html.
Computer scientist Elizabeth Bradley has a Web site at http://www.cs.colorado.edu/~lizb/Home.html.
Comments are welcome. Please send messages to Ivars Peterson at email@example.com.