Sounds of the Mandelbrot Set

Source: Sounds of the Mandelbrot Set, CodeParade, 9:32, uploaded 2021-02-28, category Education, playlist index 1623.

CodeParade starts with a small irritation. Fractals already look interesting, and some of them have found their way into video games, but their sound remains unexplored. A quick prototype turns the Mandelbrot set into an instrument, then tests the idea against other fractals whose orbits carry more unstable behaviour.

From black pixels to orbits

The Mandelbrot set assigns a point $c$ on the complex plane to a simple iteration. The programme copies $c$ into $z$, updates $z$ over and over, and watches what happens. A point whose orbit drifts away towards infinity sits outside the set and receives a colour. A point whose orbit converges, settles into a repeating cycle, or behaves in another bounded way belongs to the set and appears black.

The complex-number notation keeps the rule short. Ordinary horizontal and vertical coordinates would work as well, although the equation would look less tidy. The more useful detail lies in the path each point takes. One starting point may converge to a single value. Another may settle into an orbit with period three, four, or five. Other starting points produce different shapes before they escape.

The usual image throws most of that behaviour away. Every bounded orbit becomes the same black pixel, even though the paths inside those pixels have different periods and forms. CodeParade treats the orbit as a signal instead. The Mandelbrot picture then becomes a map of possible sounds.

Coordinates as a stereo signal

The first instrument uses the point’s $x$ and $y$ coordinates as the amplitudes of the left and right speakers. CodeParade samples the result at 8 kHz, then interpolates it to the more familiar 48 kHz playback rate. The low sampling rate keeps the prototype cheap enough to run on the GPU in real time, while interpolation smooths the output for listening.

Repeating orbits produce repeating waveforms. Their periods become audible as fundamental frequencies, so the sound gives a direct sense of the orbit’s cycle. Zooming into a bulb adds another harmonic on top of the original sound. Moving between bulbs therefore changes the timbre as well as the pitch.

The first tones are too clean to be pleasant for long. Dampening gives them a decay and makes the result feel more like an instrument. The patch remains playable, although the Mandelbrot set contains a serious limit for this experiment. Its most chaotic regions are unstable. Reaching one of them would require infinite precision, so every finite click eventually lands on an orbit that settles into a repeating pattern. The resulting sound stays a combination of pure tones.

That limit changes the next question. The instrument can expose the Mandelbrot set’s periods, yet it cannot reach the unstable behaviour that gives chaos its roughness. CodeParade turns to fractals with stable chaotic regions.

The Burning Ship and stable chaos

The Burning Ship fractal gets its name from its shape. Its useful property here is mathematical. Some of its chaotic regions remain stable enough to select, which produces sounds with a much rougher and more unsettling character than the Mandelbrot tones.

CodeParade then colours the fractal according to the orbit rather than leaving the bounded area black. The new view separates regions that converge to a point from regions that converge to cycles, whilst also showing different forms of chaotic behaviour. The colouring gives the sound a visual explanation: areas that look similar in the ordinary set can contain different orbit structures and therefore produce different audio.

The Feather fractal as an infinite piano

CodeParade’s own Feather fractal provides a different kind of instrument. It contains many good clusters, and each cluster supplies a different note. Zooming through the set feels like searching through an infinite piano whose notes keep acquiring new tonal relations at smaller scales.

The visual form matters as well. The Feather fractal has a delicate, repeated structure that makes the map easy to explore. Its appeal comes from the same relationship that made the Mandelbrot set useful: the image gives a place to move, while the orbit supplies the sound attached to each place.

The sound-effects fractal

The next example reverses the visual appeal. CodeParade calls it the sound-effects fractal because it looks plain yet produces a wide range of effects. The orbits have a strong symmetry, and colouring those orbits exposes a complicated internal structure. That structure gives the fractal more sonic variety than its surface suggests.

The result recalls the small sound generators used in game jams, especially tools such as SFXR. Those programmes make short effects through compact combinations of oscillators, noise, envelopes, and pitch changes. CodeParade suggests that this fractal could serve a similar purpose, with the orbit acting as the source of the variation. The video presents that as a useful possibility rather than a tested production workflow.

Julia sets and the loss of variety

Julia sets seem like the obvious next family to explore. CodeParade leaves them until late because ordinary Julia sets tend to sound dull. For a point on a Julia set, the orbit converges to the same orbit associated with the corresponding point in the Mandelbrot set. The visual boundary changes, while the available audio often repeats the same underlying behaviour.

Some variants escape that restriction. Certain Julia sets of the Burning Ship fractal are bistable. A single set can converge to one of two orbits, depending on the starting point, so the same visible object contains two possible outcomes. More exotic chaotic maps can go further. CodeParade names the Chirikov map as an example whose Julia orbits form a continuum rather than collapsing to a small set of repeating choices.

The distinction gives the project a useful rule of thumb. A fractal becomes an interesting instrument when its orbit structure supplies meaningful variation that the image alone hides. Visual complexity by itself says little about the range of sounds.

Limits of the demonstration

The video is a rapid experiment built around a working prototype. It explains the Mandelbrot iteration, gives the stereo mapping and sampling rates, and compares several fractal families through short demonstrations. CodeParade describes the sounds as creepy, beautiful, or varied. Those judgements belong to the demonstrations themselves, so a text note can preserve the selection and the mechanism without reproducing their timbre.

The source also leaves several engineering details open. It does not specify the interpolation method, the exact dampening curve, the GPU precision, or the implementation of the Burning Ship, Feather, and sound-effects variants. It gives no measurements of pitch, harmonic content, or perceptual similarity between locations. The project shows that fractal orbits can drive an exploratory instrument. It does not establish a general method for composing or synthesising sound from every fractal.

Further reading / references

• Fractal Sound Explorer, the playable experiment linked in the video description.
• FractalSoundExplorer source code, the GitHub repository linked in the video description.