Broader Impacts is a computer-generated musical composition constructed from the mathematics of aperiodic tilings. It reimagines rhythm and pitch not as repetitive, grid-bound elements, but as the outcome of recursive and hierarchical rules—rules that generate structure without repetition.
At its foundation is a substitution system (sometimes known as an L-system) involving four symbols: A, B, C, and D. Each of these symbols expands into a five-letter string of the same alphabet:
- A → AADBC
- B → BABDC
- C → CDABB
- D → DADBC
This rule is applied recursively: starting with a single letter, one obtains longer and longer strings that encode increasingly complex layers of temporal organization. The substitution acts like a musical fractal—self-similar across levels, but never exactly repeating.
These symbols aren’t just structural—they also produce sound. Each symbol is assigned a pitch, but the specific pitch depends on the hierarchical level of the substitution tiling in which it appears. The result is a system with twelve distinct pitches: four base pitches (associated to A, B, C, and D), four deeper pitches for the next level of substitution, and four more for the level beneath that. This structure allows the same symbolic material to be interpreted differently depending on its depth in the compositional hierarchy, producing a layered and resonant sonic texture.
The timing of the notes is derived from the same substitution structure. Each symbol corresponds to a tile of a certain length—these lengths are determined by solving a system of linear equations based on the substitution rules. The result is a rhythm that is mathematically coherent and hierarchical, but fundamentally aperiodic: it does not loop, pulse, or resolve in the way traditional meters do.
Broader Impacts is therefore a composition that emerges entirely from the logic of the substitution system. It translates a recursive symbolic structure into musical time and pitch, creating a piece that is at once structured and surprising, formal and expressive.
While not written with human performance in mind, the piece is rendered electronically to faithfully reflect the intricate timing and pitch relationships. It stands as both a musical work and a conceptual demonstration: a model for how mathematical ideas—especially those drawn from geometry and algebra—can shape musical form in radical new ways.
Broader Impacts received its world premiere on April 26, 2025, at Petworth Porchfest. It was conducted by an Arduino Mega 2560 Rev3, with support from MATLAB. For more background on the ideas behind the piece, see Rodrigo Treviño’s paper Quasimusic, where Broader Impacts appears as the first recurring example to illustrate the theory in sound.