From the start, it was clearly evident that the melodic content of my composition would need to be derived from the visual content of the Lorenz attractor. Gil Goldstein's Jazz Composer's Companion, a treasury of creative musical concepts, explores the idea of utilizing graphs as a depiction of melodic lines.
Graphs give us a kind of x-ray look at the nature of melodies by exposing important characteristics, such as the contour, rhythmic design, and range of a melody instantly.
Although graphing the entire Lorenz attractor in three dimensional space wouldn't make any sense as far as a melodic line was concerned, graphing a single one of its variables versus time would.
Figure: Lorenz Attractor X variable vs. Time
And in order to avoid chromatic scale runs as figure 4.1 might necessitate, I quantized the entire graph, taking data from every fifth time step (figure 4.2).
Figure 4.2: Quantized Lorenz Attractor X variable vs. Time
One of the most intriguing side effects of chaos, as stated previously, was sensitive dependence on initial conditions. As I wished to re-create the mathematical chaos in some capacity as musical chaos, this effect became a unique compositional tool. By running the simulation with an experimentally obtained, slightly different set of initial conditions, I could generate two melodic lines that `tracked' each other at the start, but would eventually degrade to the point where no correlation could be observed between the two (figure 4.3).
Figure 4.3: Chaos at work
The result often had both horn players playing slightly out of sync - a technique very similar to one which Ornette Coleman applied in his own compositions. Both he and Don Cherry would think of starting on the first note at the same time, hitting every one in between, and ending together on the last note; what happened in between was up to the performance.
One compositional change that I saw fit to make was in the second ending for the A section of the piece. As an attempt at a certain kind of symmetry, I used the two bars from the first ending and intervallicly inverted them to create the second ending.
All of the graphs used in the composition are presented in a more readable `stretched' format in Appendix E.