Metamathematics of Algorithmic Composition

TL;DR

The work investigates the fundamental limits and possibilities of algorithmic music composition through the lens of computability and complexity theory, using fractal dynamics from the Mandelbrot/Julia family and IFS as central exemplars. It analyzes finitary versus infinitary computational methods, highlighting irreducibility and uncomputability as core barriers, while arguing that parametric maps can offer powerful insights if tractable, particularly under a hypothetical $\,\mathsf{P}=\mathsf{NP}$ regime. The author surveys toolkit-based composition, intrinsic vs extrinsic algorithms, live coding, and AI approaches (notably LLMs), weighing their musical expressiveness, opacity, and practicality. The paper concludes that humans remain essential for guiding algorithmic processes, and that advancements in hardware, software tooling, and principled geometric mapping of musical possibilities are key paths forward for the field.

Abstract

This essay recounts my personal journey towards a deeper understanding of the mathematical foundations of algorithmic music composition. I do not spend much time on specific mathematical algorithms used by composers; rather, I focus on general issues such as fundamental limits and possibilities, by analogy with metalogic, metamathematics, and computability theory. I discuss implications from these foundations for the future of algorithmic composition.