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 acousticwave equation and modal superposition, we derive resonances of strings and tubes and modelensemble performances as convolutional networks of linear time-invariant (LTI) and weaklynonlinear filters. We formalize pitch-class structure with group actions on Z12, represent scoresas Markov processes with memory, and introduce a stochastic, monoidal category of denotatorsthat composes score → gestural performance → perception. A spatial case study of Corigliano’sCircus Maximus demonstrates how the theory predicts spectral/spatial energy flows across threewind/percussion sub-ensembles. The framework is constructive: it yields estimators for transitionpriors and functorial morphisms that are learnable from recordings and symbolic corpora, and ithighlights testable predictions for perception. This article develops, corrects, and extends ideassketched in a recent presentation.