for harp and double bass

*Implied Chaos* is a musical depiction of a fractal, a mathematical pattern where each part is a miniature image of the whole. This fractal, known as the logistic map, is itself a graphical display of the logistic growth function, an equation which governs the rate of growth of a population bounded by some upper limit, such as an ecosystem with limited resources. Depending on the parameter of how fast the population reproduces relative to the maximum capacity of the environment (represented by the horizontal axis of the image), the actual level of the population (on the vertical axis) may reach a stable equilibrium, oscillate between high and low levels, or explode and crash wildly. In particular, very small differences in the parameter may mean the difference between a regular cycle of “boom” and “bust” years or a chaotic series of levels with no apparent rhyme or reason.

In 1975, mathematician James Yorke proved that any logistical system that displays such regular cycles with odd periods (repeating every 3, 5, or 7 years) is also susceptible to chaotic behavior, a stunning discovery which essentially launched the mathematical field now known as chaos theory. The title of Yorke’s paper was simply *Period Three Implies Chaos*.

In this piece, the musical gestures undergo gradual transformation from order to chaos in all aspects, including pitch, timbre, and texture. At first, the sounds are so pure they can barely be heard. Gradually, they coalesce into a low C which immediately begins rising in pitch. As its rise slows, it splits into two voices, then into four, and so on until reaching chaos, all the while becoming noisier and noisier. Yet, in the middle of the most chaotic regions, like the eye of a hurricane, appear the “windows of order” described by Yorke, in which a regular ostinato gives the illusion of stability and calm, but the triple or quintuple meter ominously hints at the music’s chaotic destiny.

Musicians interested in learning about the history and theory of mathematical chaos should consult James Gleick’s fascinating book, entitled simply *Chaos*.