Strange Attractions – A Study In Tribute To Edward Lorenz

In honor of the late Edward Lorenz, I proffer the following brief insight into the basis of his life’s work… And, as a sort of token for this insight, I feel compelled to offer some of our own interpretations; interpretations, via the medium of musical/sonic form, that are derived purly from his enlightening expositions.

It is the findings and implications gleaned from Lorenz’s body of works that has so inspired me to further study their implications. As a result, I feel much the better for having glanced at the beauty behind the ‘irrational,’ which is seemingly so characteristic of certain types dynamical system. While the significance about the way various basins of attraction might interact with one another would no doubt have eventually come to light within the scientific community (something that was first alluded to by Jules Henri Poincaré), it was Edward’s work which hastened the inevitable birth of a new science proper – namely the science of non-linear dynamical systems – and so proffered to erudition a better analogy with which to understand the unpredictable eddies and flows that swirl around us all daily.

The Lorenz attractor, named after its founder, is a fractal structure corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a “3-dimensional” i.e. it has three variables, dynamical system that exhibits chaotic flow, and is well 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 for convectional flow within earth’s atmosphere. The map (see both figures 1 and 2 below) shows how the state of a dynamical system (the three variables interdependently fluxing through time within a 3-D phase-space plot) evolves in a complex, non-repeating pattern.

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 1972 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 shear complexity, along with the never ending richness of form that it generated, demonstrates the sovereign and almost “God-like” majesty that lies behind two simple basins of attraction… The resulting unpredictability woven into their subtle networks of force are truly startling!

Ever since my first glance at one of these phase-space extrapolations, I have had a deep sense of longing to port this ideal into the sonic realm so as to investigate rhythmic form, wondering whether listening to the chaos inherent within a Strange Attractor might in some way provide deeper insight into the aural aesthetics of never repeating, endless novel flow.

So without further ado, I offer a snippet from an on going project called “Strange Attractions…” A project that allows us to ‘simply’ port over some of the visual ideas seen above i.e. data flowing through 3-D phase spaces, into sonic gardens of chaotically grown rhythms and melodic modulations. While some of these recordings will no doubt sound like true chaotic flows, others will have a more musical coherence within their temporal passage. This is a direct result of some re-structuring on our part. Nonetheless, non-linear systems still reside at the core of these compositions, and thus, despite their preternatural arrangements, chaos can still be found within their flow… For it is the chaos built into the programs that we use which provides our sensibilities with the novel and diverse fragments of rhythmic undualtion from which we assimilate our sonic forms… And, just as the non-linear dynamics inherent within our body’s own biological system provides the impulse for action, we see a type cynosure functioning for our seemingly “conscious” selves, and thus the guidance is derived from that universal law of “God, or Nature.”

We hope you enjoy listening to them as much as we enjoyed making them:

Track 1 – Lorenz Experiments – Version 1.54

Kyma Strange Attractor Patterns (Pitched Metalic Spectra) by the Orange Hut studio

Track 2 – Lorenz Experiments – Version 4.692

Strange Attractions (Lorenz’s Rough Mix) by the Orange Hut studio

Track 3 – Lorenz Experiments – Version 0.78 a.k.a. Rhythm In The Numbers (Hexadecimal Mix)

Rhythm in the Numbers (Hexadecimal Mix) by the Orange Hut studio

Lastly… A big thank you to Edward Lorenz. R.I.P.