In honor of the late Edward Lorenz, I offer the following brief insight into the basis of his work while he was here with us on earth, as well as some of my own interpretations of his ideals in musical form.
Picture 1. Edward Norton Lorenz - May 23, 1917 to April 16, 2008
It is the findings and implications gleaned from within Lorenz’s body of works that has so richly inspired me to study further the implications and beauty behind the chaos inherent in all dynamical systems. While the implications about the way basins of attraction interact with one another would no doubt have eventually come to light, it was Edward’s work that hastened the inevitable birth of a new science; namely that of chaos, and so proffered to science to better analogy with which to understand the unpredictable eddies and flows that swirl around us in nature and life daily.
The Lorenz attractor, named after Edward Lorenz, is a fractal structure corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its ‘lemniscate’ shape (a term used in algebraic geometry to refer to an object that has a likeness to the figure eight’s form).
The oscillator was originally used by Lorenz as a simplified model of convection flow in the earths atmosphere. The map (see both figure 1 and 2 below) shows how the state of a dynamical system (three variables varying in time plotted in a 3-D phase-space) evolves in a complex, non-repeating pattern.
Figure 1.
Figure 1. appeared in the Nature journal 31 August 2000, pp 949 as part of an article titled The Lorenz Attractor Exists, written by Ian Stewart. It was created as part of an OpenGL interactive viewer and rendered on a farm of Dec Alphas using ProRay.
In 1961, Lorenz had managed to create a skeleton of a weather system from a handful of differential equations. He kept a continuous simulation running on an extremely primitive analog computer that would output a day’s progress in the simulation every minute as a line of text on a roll of paper. Evidently, the whole system was very successful at producing “weather-like” output – nothing ever happened the same way twice, but there was an underlying order that delighted Lorenz and his associates.
…Line by line, the winds and temperatures in Lorenz’s printouts seemed to behave in a recognizable earthly way. They matched his cherished intuition about the weather, his sense that it repeated itself, displaying familiar patterns over time, pressure rising and falling, the airstream swinging north and south. (GLEICK, J. Chaos: Making a New Science.)
What Edward Lorenz had discovered was a chaotic system. Even though a computer had control of the simulation, and certainly possessed the capability to generate random numbers at will, there was nothing random about any portion of the way the simulation was supposed to work. It merely followed the laws of calculus as set down by Sir Isaac Newton himself and outputted a day’s worth of virtual weather at the end of each minute. Lorenz’s initial brush with chaos is described best by James Gleick’s own words, from Chaos:
One day in the winter of 1961, wanting to examine one sequence at greater length, Lorenz took a shortcut. Instead of starting the whole run over, he started midway through. To give the machine its initial conditions, he typed the numbers straight from the earlier printout. Then he walked down the hall to get away from the noise and drink a cup of coffee. When he returned an hour later, he saw something unexpected, something that planted a seed for a new science.This new run should have exactly duplicated the old. Lorenz had copied the numbers into the machine himself. The program had not changed. Yet as he stared at the new printout, Lorenz saw his weather diverging so rapidly from the pattern of the last run that, within just a few months, all resemblance had disappeared. He looked at one set of numbers, then back at the other. He might as well have chosen two random weathers out of a hat. His first thought was that another vacuum tube had gone bad.
Suddenly he realized the truth. There had been no malfunction. The problem lay in the numbers he had typed. In the computer’s memory, six decimal places were stored: .506127. On the printout to save space, just three appeared: .506. Lorenz had entered the shorter, rounded-off numbers, assuming that the difference-one part in a thousand-was inconsequential.
It was a reasonable assumption. If a weather satellite can read ocean-surface temperature to within one part in a thousand, its operators consider themselves lucky. Lorenz’s Royal McBee was implementing the classical program. It used a purely deterministic system of equations. Given a particular starting point, the weather would unfold exactly the same way each time. Given a slightly different starting point, the weather should unfold in a slightly different way. A small numerical error was like a small puff of wind – surely the small puffs faded or canceled each other out before they could change important, large-scale features of the weather. Yet in Lorenz’s particular system of equations, small errors proved catastrophic.
And there is the show-stopper: small errors prove catastrophic. Lorenz entitled a 1979 paper, “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” and the title stuck. Today, sensitive dependence on initial conditions is referred to as “The Butterfly Effect.”
For the purposes of experimentation, Lorenz created a new system with three nonlinear differential equations:
dx / dt = a (y – x)
dy / dt = x (b – z) – y
dz / dt = xy – c z
It was a reduced model of convection, similar to the swirls of cream in a hot cup of coffee, only much, much, much simpler. And yet the complexity and never ending richness that it generates within a phase-space, demonstrating the sovereign and noble interplay behind the basins of attraction and the unpredictability behind their subtle networks of force are startling!
Ever since my first glance of one of these phase-space extrapolations of the relationships between the 3 variables in phase-space, I have had a deep sense of longing to port this ideal into the sonic realm of rhythmic form, wondering whether listening to its chaotic nature might in some way provide further insight into the aural aesthetics of never repeating, endless flow.
Figure 2.
So without further ado, I offer a snippet from an on going project called “Strange Attractions”. A project that allows me to ’simply’ port over the visual ideas seen above i.e. data flowing through 3-D phase space, into a sonic garden of chaotically grown rhythms and rhythmic modulations… For it is the chaos inherent within the programs that guide the composition of their own rhythmic undualtions, just as we humans guide ourselves from the chaos inherent within our minds. We hope you enjoy listening to them:
Track 1 – Lorenz Experiments – Version 1.54
Track 2 – Lorenz Experiments – Version 4.692
Track 3 – Lorenz Experiments – Version 0.78 a.k.a. Rhythm In The Numbers (Hexadecimal Mix)
Lastly… A big thank you to Edward Lorenz. R.I.P.
