My project is an attempt at synthesizing sound from chaos. The final video I created shows two simulations of nearly identical double pendulums. The double pendulum system is one of the simplest ways to show chaotic motion. To quote Edward Lorenz, chaos is “when the present determines the future, but the approximate present does not approximately determine the future.” To the naked eye, both simulations appear to start out the same and one would think that the pendulums would follow the same path. In reality, the bottom simulation’s the blue mass starts out slightly further to the right. This small difference in initial conditions gives us a good way to visualize a chaotic system as it evolves. After a period of time, the paths taken by each simulation become drastically different. My goal for this project is to sonify the effects of this chaotic system.
I was inspired to do this through my study of chaotic motion in my classical mechanics course, and by David Tudor’s ‘Bandoneon!’ In Bandoneon!, Tudor attempted to create a “rebirth from white noise” by using a large array of electroacoustic devices and feedback system. In my project, I am attempting to creating a rebirth from the “white noise” of the diverging paths the pendulums take. Rather than generating sound from the similarities between the simulations, I based all my sounds around how the positions and velocities of the two simulations differed. This creates a piece that starts off simple and evolves over time, reflecting the chaotic nature of the pendulums.
All the simulations and sound synthesis were done in a program called Mathematica. I started off by generating a complex waveform composed of many sine and cosine waves whose frequencies and amplitudes depend on the differences in position and velocity of the two simulations. I used equations for intermodulation and vibrato to represent chaos. Intermodulation takes two waveforms, adds them, and squares that sum to generate a series of waveforms that contain unique frequencies. One of the mathematical prerequisites for chaos is that the system must be nonlinear, and intermodulation is a nonlinear operation. This means that more chaotic processing is used in the production of these complex tones. The vibrato is used to soften these rather harsh tones and attempt to draw in the listener. Because all these tones were generated using the differences between the double pendulum simulations, they start off almost nonexistent. As the piece progresses, the pendulums begin to differ more and more and the sound gets more and more complex.
Running the simulation at full speed produced a waveform that changed extremely fast and was not very listenable. To ameliorate this issue I slowed the piece down eight times, which created a waveform that was easier to follow and listen to. With vibrato added and the piece slowed down, the piece still felt harsh and “noisy” as the sounds became increasingly complex. I decided to work with Mathematica’s built in instrumentation to take my compilation of complex tones from a sound to a piece of sound art.
I mapped the differences in position and velocity of the four masses onto three instruments. Each decimal value was pushed to the nearest integer to create clean and western intervalled tones.The positions (horizontal and vertical) were all mapped to a choir of voices singing “Ooo,” the velocities (horizontal and vertical) to a group of pan flutes, and the vertical positions and vertical velocities to a quartet of violins. I chose these three instruments because they blend together well and overcome the artificial feeling most MIDI instruments give off. The tones played by each set of instruments evolve over time, much in the same way the complex waveform does. Each set instruments start off playing a single note. As the pendulums’ paths begin to diverge, there are small fluctuations in pitch, and eventually the instruments become a symphonic cacophony. The tempo of the piece remains constant throughout. This is done to keep the listener’s interest. Keeping a steady tempo is the best way to create music that draws in a listener, and chaotic tempo is anything but steady. Thus my piece sacrifices a pure representation of chaos in both pitch and tempo so that the listener has something to grasp onto amidst all the chaos.
After synthesizing all my sounds, I worked in Reaper to map chaos onto the panning of the piece. I had four tracks (one for the complex tone, one for each set of instruments) and I used the horizontal and vertical positions and velocities to control how each track panned. If the difference in positions of a mass between the two simulations was greater than one, I looked at the velocity of that mass at that time and multiplied it by ten to get the percent I would pan. Positive velocities meant left pan and negative meant right pan. The result was, you guessed it, rather chaotic panning. This was exactly what I wanted though, and it helps add more chaos to the piece without being off putting the the listener.
After a couple more cosmetic edits in Reaper, I worked on syncing up my sound with my animations. My final piece stands as a combination of visual and auditory stimuli. The animations and audio give a visual and sonic depiction of the pendulums and how they diverge. My final piece uses the animations as a guide for the ear to listen to the sounds of chaos. Please look, listen, and enjoy my piece!