Gestural Dynamics in Modulation: (Towards) a Musical String Theory

We prove a modulation theorem for diatonic scales that is based on the theory of hypergestures and vector fields derived from inner symmetries of diatonic scales and Lie bracket fields. It yields the same modulation degrees as the classical model (Mazzola, Gruppen und Kategorien in der Musik, 1985, [1], Geometrie der Töne, 1990, [2], Mazzola et al., The Topos of Music-Geometric Logic of Concepts, Theory, and Performance, 2002, [3]), which confirmed Schoenberg’s modulation theory (Schoenberg, Harmonielehre 1911, Universal Edition, Wien 1966, [4]). In this hypergestural model, integration of differential forms is considered. In this context, we can model and prove Stokes’ theorem for hypergestures, generalizing the classical case. Stokes’ theorem is a central result in differential geometry, relating the integral of the derivative of a form to the boundary of its domain of integration. It has important application in physics, such as in mechanics (integral invariants, see (Abraham, Foundations of Mechanics, 1967, [5])) or in electrodynamics (relating differential and integral forms of Maxwell’s equations (Jackson, Classical Electrodynamics, 1998, [6])). The basic form of this theorem deals with integration on singular hypercubes. In (Mazzola, J Math Music 6(1):49–60, 2012, [7]) we have extended singular homology on hypercubes to singular homology on hypergestures. It was therefore straightforward to try to extend Stokes’ theorem to hypergestures.