Musical tone coloring via bifurcation control of Eulerian n-tuple Hopf singularities

TL;DR

This paper models musical tone coloring through bifurcation control of Eulerian systems with Hopf singularities, enabling precise spectral and temporal envelope synthesis for musical notes like C#4 from piano and violin recordings.

Contribution

It introduces a novel dynamical modeling approach using bifurcation control of Eulerian n-tuple Hopf singularities for musical tone coloring, linking spectral envelopes with bifurcation scenarios.

Findings

Accurately constructs spectral and amplitude envelopes of musical notes.

Demonstrates bifurcation control on real piano and violin recordings.

Reveals complex bifurcation scenarios including Clifford toral manifolds.

Abstract

An intrinsic essence of sounds in music is the evolution of their qualitative types while in mathematics we interpret each qualitative change by a bifurcation. Hopf bifurcation is an important venue to generate a signal with an arbitrary frequency. Hence, the investigations of musical sounds via bifurcation control theory are long-overdue and natural contributions. In this paper, we address the tone coloring of sounds by dynamical modeling of spectral and temporal envelopes. Multiple number of leading harmonic partials of a note (modulo a hearing sound velocity threshold) are attributed into an Eulerian differential system with n-tuple Hopf singularity. The qualitative evolution of the temporal envelop is then simulated over a set of consecutive time-intervals via bifurcation control of the differential system. For an instance, our proposed approach is applied on audio C#4 files obtained from piano and violin. Fourier analysis is used to generate the amplitude spectral vectors. Then, we associate each amplitude spectral vector with an Eulerian flow-invariant leaf. Bifurcation control suffices to accurately construct the desired spectral and amplitude envelopes of musical notes. These correspond with a rich bifurcation scenarios involving Clifford toral manifolds for the Eulerian differential system. In order to reduce the technicalities, we employ several reduction techniques and use one bifurcation parameter. We show how different ordered sets of elementary bifurcations such as pitchfork and (double) saddle-node bifurcations are associated with the qualitative temporal envelop changes of a C]4 played by either a piano or a violin. A complete hysteresis type cycle is observed within the temporal envelop bifurcations of the C#4 played by violin.