A dynamical system is a system whose state evolves with time over a state space according to a fixed rule. A dynamical system is all about the evolution of something over time. To create a dynamical system we simply need to decide (1), what is the “something” that will evolve over time and (2), what is the rule that specifies how that something evolves with time… A dynamical system is simply a model describing the temporal evolution of a system.
There is a rich history of music inspired by and connected to mathematical and scientific notions. I first became fascinated by dynamical systems because there seemed to be so many ways they could relate to art. They can form extra-musical abstractions which inspire the composition of a piece of music, where a composer attempts to embody the qualities of dynamical systems in entirely poetic ways; they can model composition techniques which utilize semi-autonomous processes, taking the trajectory of the music out of the composer’s hands. They can galvanize both algorithmic and generative composers who create electronic works based on number systems, and sound installation artists developing physical works that can run autonomously. This is a playlist of pieces and installations influenced by, embodying, and which may even be dynamical systems.
Zimoun, “1944 prepared dc-motors, mdf panels 72×72 cm, metal discs Ø 8 cm” (2020)
Broadly speaking, a dynamical system is a mathematical model that can be used to understand the behaviors of systems. A dynamical system changes and evolves over time, according to a fixed set of rules: Think of planetary bodies, whose movement is determined by the laws of gravity, or of a pendulum swinging back and forth until it runs out of energy. These are examples of regular, simple systems which show stable behavior and cycle through the same set of states.
Zimoun is a Swiss artist who makes amazing physical installations: globally predictable but locally unpredictable mechanical systems designed to sonically activate the acoustics of the space they are in. If the energy never died you could come back in 100 years and it would still be doing the same things, although the precise relative positions of each component would not be repeated.
Alvin Lucier, “Music on a Long Thin Wire” (1977)
Dynamical systems become particularly interesting when you introduce chaos. Chaos is essentially the scientific word for unpredictability. Edward Lorenz summarizes chaos as “when the present determines the future, but the approximate present does not approximately determine the future.” Some dynamical systems can become chaotic easily because of their “sensitivity to initial conditions.” All you need to do to make a pendulum act chaotically is add a joint to create a double pendulum: the trajectory of the second pendulum will be different every single time and entirely unpredictable, because the exact position that the pendulum is dropped from can never be exactly replicated. However small the change of angle is, it will permeate through the system.
Alvin Lucier‘s “Music on a Long Thin Wire” is an unpredictable, chaotic system with behaviors and sounds that emerge differently every time. Whereas Zimoun contains his chaos within careful constructed physical boundaries, Lucier actively embraces opening his installation to outside influences. “Fatigue, air currents, heating and cooling, even human proximity could cause the wire to undergo enormous changes,” he writes. “Footsteps on the floor caused extremely slight shifts in the positions of the tables to which the wire was clamped, causing spectacular changes in the sound of the wire. [Someone] who slept overnight under the wire reported that even with no movement in the room it would mysteriously erupt into triadic harmonies…”
Tristan Murail, “Attracteurs étranges” (1992)
The fairly well-known graphic of the Lorenz attractor consists of the plot points of chaotic systems—the butterfly shape is created as the line never repeats the exact same trajectory. The lines revolve around certain fixed points (a set of states around which a system will evolve) known as “attractors.” It’s an abstract representation of a naturally chaotic system, where the slightest variation can lead to huge effects later on. The saying “the flap of a butterfly’s wings in Brazil may cause a tornado in Texas months later” encapsulates this idea.
In his piece “Attracteurs étranges,” Tristan Murail attempts to embody these phenomena in what he describes as a “poetically analogous” way. “The melodic contours trace spirals with trajectories that always seem to return toward one or several of the same points, but in fact often follow varied, distorted or twisted paths,” he writes. “We sometimes seem to reach a point of equilibrium: but the equilibrium is unstable and projects the music into a new cycle of oscillations.”
Steve Reich, “Marimba Phase” (1967)
The ‘60s minimalists are heavily associated with process music, and if the definition of a dynamical system is taken as noted above, then Steve Reich’s phase pieces serve as a great musical analogy for them. These phase pieces consist of two musical parts: one part gradually speeds up, creating an element of autonomy in the resulting sound output. The composer’s hand is seen in the chosen musical materials, here creating a simple, linear system with a musical output which is entirely predictable, though no less musically enjoyable for it. In “Music as a Gradual Process,” Reich writes, “I am interested in perceptible processes. I want to be able to hear the process happening throughout the sounding music… Though I may have the pleasure of discovering musical processes and composing the musical material to run through them, once the process is set up and loaded it runs by itself.”
György Ligeti, “Clocks and Clouds” (1973)
In 1966, philosopher Karl Popper wrote a paper titled “Of Clouds and Clocks: An Approach to the Problem of Rationality and the Freedom of Man.” This paper was an early response to reductionism in science, which for a long time stopped people trying to even understand chaos and unpredictably. Popper argued that the mindset of many scientists was to reduce a problem until it became solvable and ignore the bits that make a problem too difficult: “Science may be described as the art of systematic oversimplification—the art of discerning what we may with advantage omit.”
Popper writes of a juxtaposition in his essay, stating: “My clouds are intended to represent physical systems which, like gasses, are highly irregular, disorderly, and more or less unpredictable… on the other extreme we may place a very reliable pendulum clock, a precision clock, intended to represent physical systems which are regular, orderly, and highly predictable in their behavior.” Ligeti was clearly inspired by dynamical systems and how they appear in nature, attempting to musically mimic the clocks and clouds of natural systems with intricately written musical lines which—though incredibly precise on their own—create cloud-like textures when in combination with one another. A form of art imitating life and science.
Robin Meier and Andre Gwerder, “Synchronicity (Thailand)” (2015)
A violin string’s partials will be slightly out of tune for the briefest of moments until they phase lock and become exact integer multiples of the fundamental frequency. Put metronomes on a wooden board and they will synchronize due to sympathetic vibrations. In this experiment by Robin Meier & Andre Gwerder, thousands of live fireflies are made to synchronize their flashes with computer-controlled LEDs. Phase locking is a phenomenon of dynamical systems which literally holds the universe together.
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Alan Lamb and Chris Watson, BBC4 Interview (2014)
Wire fencing in nature vibrates and makes sound just like any other vibrating string, though perhaps not in quite as audible ways. The right equipment, approach and ear can reveal the most incredible resonances. This is perhaps the purest form of dynamical systems music-making: remove yourself altogether and record what is already there.
Rolf Wallin, “Stonewave” (2021)
Rolf Wallin’s “Stonewave” is a great example of composition techniques inspired by fractals. Wallin musically captures the essence of dynamical systems and their connection to the organicism of the natural world with this piece. “The last few years I have become increasingly involved in some peculiar mathematical formulas called fractals,” Wallin writes. “These formulas, used in the fast-growing field of chaos theory, are relatively simple, but they generate fascinating and surprisingly ‘organic’ patterns when shown graphically on a computer screen, or played as music.”
Horatiu Radulescu, “Das Andere” (1984)
Emergence occurs when something becomes more than the sum of its parts. “Thanks to the interactions between the components of a complex system, yet another typical feature of complex dynamic systems arises, which is emergence,” linguist Paul Van Geert explains. “Emergence means that as a consequence of the interactions between components, the interacting whole acquires or creates new properties, i.e., properties that cannot be reduced to properties of the components or to simple additions of those components.” Although I doubt Horatiu Radulescu was thinking of dynamical systems when he composed the masterpiece “Das Andere,” the aural effect created by this sonic sound bath of natural string resonances and intense bowing technique fits the description well.
Scott McLaughlin, “The Endless Mobility of Listening” (2015)
Scott McLaughlin explores the emergence of indeterminate string partials in this longform drone bowing piece. It cannot be accurately predicted when or which partials will “pop out” with this kind of bowing technique, but when they do, the violinist uses a foot pedal to electronically loop the partials, building a kind of spectral sound collage on fly.
Ryoji Ikeda, Live at Sou Fesvtial (2017)
Computer programs such as Max/MSP and Pure Data can be used to create semi-autonomous algorithmic musical systems. One of the best known composers to use these techniques—alongside the sonification and visualization of scientific/mathematical data—is Ryoji Ikeda.
Emily Howard, “Leviathan” (2016)
Emily Howard created this dynamical systems-inspired piece in response to data provided by Professor Lasse Rempe. Rempe says, “No matter how chaotic a system may be, there are always also some regular systems nearby.” This piece plays with this idea on musical and rhythmic level, creating a sense of stability with finely drawn out phrase lengths which may appear improvised, but are actually composed to the finest detail. ¶
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