April 25 2007

MAIN POINTS

Sarah defined chaos as behavior that is aperiodic and unpredictable over the long-term, yet deterministic (i.e, based on differential equations), and very sensitive on initial conditions. In 3-Space, for instance, a starting point of (0.99, 1, 1) will have very different long-term behavior than (1, 1, 1), with the difference between the two models growing exponentially as t increases. We saw this in our last lab. The Lorentz equations were derived to model weather systems, so this is evidence for how only close forecasts of the weather are accurate, because small errors in measurements of the initial condition quickly snowball out of control. Sarah used this sensitivity in composing music. She mapped a piano piece to a pitch sequence, and assigned that pitch sequence onto an existing strange attractor (I'm assuming it was the attractor starting at (1,1,1) with h=0.01, for instance). Then she chose a very similar starting point, and created the piece by approximating the differential equation at each discrete step t using the Runge-Kutta numerical method. The new piece started similarly to the old piece, but quickly deviated. However, it maintained some of the same tonal qualities. Due to the limitations of the model, rhythm stayed the same, in discrete 8th notes or whatever they were. It sounded great and the talk was packed with music people. CHALLENGES I didn't quite see how she was able to map the pitch sequences onto the Lorentz attractor, and then shift the notes based on the new attractor... She must have carefully chosen the mapping to make the new piece have similar key and chords, because I would think that the key would start to jump all over the place. I'd like to know how many pieces she created before she came across the one she settled on. REFLECTIONS The project raises some questions— when a composer creates variations on a theme, how are these variations different? What is the creative process there? Can chaos be used to model this creative process? This project sidestepped probabilistic methods entirely and used deterministic behavior built into the Lorentz equations instead.