From Acoustic Mechanics to a Category of Perceptual Music: A Mathematical Framework with a Spatial Case Study of Corigliano's Circus Maximus

We present a unified mathematical account of musical sound that links mechanistic acoustics,
score structure, and human auditory perception. Starting from the three-dimensional acoustic
wave equation and modal superposition, we derive resonances of strings and tubes and model
ensemble performances as convolutional networks of linear time-invariant (LTI) and weakly
nonlinear filters. We formalize pitch-class structure with group actions on Z12, represent scores
as Markov processes with memory, and introduce a stochastic, monoidal category of denotators
that composes score → gestural performance → perception. A spatial case study of Corigliano’s
Circus Maximus demonstrates how the theory predicts spectral/spatial energy flows across three
wind/percussion sub-ensembles. The framework is constructive: it yields estimators for transition
priors and functorial morphisms that are learnable from recordings and symbolic corpora, and it
highlights testable predictions for perception. This article develops, corrects, and extends ideas
sketched in a recent presentation.