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|Paper IPM / M / 16728||
Qualitative changes in sound intensities frequently occur in musical signals and Hopf bifurcation control is the most natural mathematical approach to study them. We 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 (a three-parametric family) 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. We apply our proposed method to the first three bars of a sheet music. This is implemented in MATLAB so that one can hear the smooth dynamic effect of bifurcations using a computer through a classical sheet music. Then, one can hear how pitchfork (subcritical and supercritical) and double fold-bifurcations of invariant hypertori sound in music.
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