Bifurcation control and sound intensities in musical art (Q2048498)

The authors propose a natural formulation for investigating and generating varieties of pitch simultaneity and sound intensity using Eulerian flows with \(n\)-tuple Hopf singularity for a sufficiently large \(n\). Chord leaves and chord limit sets are defined as two critical flow invariant manifolds. The reduction of flows on chord leaves facilitate the study and bifurcation control through a generic codimension-three scalar bifurcation problem. There is a one to one correspondence between bifurcations of positive equilibria in the scalar equation and bifurcations of invariant chord limit sets. An ordered set of double saddle-node and pitchfork bifurcation control of subcritical and supercritical types can describe and generate the sound intensity bifurcations of a sheet music. These scalar bifurcations are interpreted in terms of appearances and disappearances of various chord leaf-stable and unstable flow-invariant chord limit sets. This topic is interesting and the obtained results are completely novel. This work displays an important application in music.