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Paper   IPM / M / 17116
School of Mathematics
  Title:   Tone colour in music and bifurcation control
  Author(s):  Majid Gazor (Joint with A. Shighi)
  Status:   Published
  Journal: Journal of Differential Equations
  Vol.:  326
  Year:  2022
  Pages:   129-163
  Supported by:  IPM
An intrinsic characteristic of musical sounds is the qualitative evolution of their timbre (tone colour). In this paper, we interpret each qualitative changeby a bifurcation. We employ spectral and temporal envelopes as two of the most salient features of tone colour. Fourier analysis of a note generates its n-leading harmonic partials and an amplitude spectral n-vector A for n ≥2. We introduce harmonic manifolds associated with amplitude spectral vectors. This is to accommodate the n-leading harmonic partials of the note. Then, harmonic partials are modelled by solutions of a radial coupling of n-tuple parametric Hopf oscillators. We use one bifurcation parameter and interpolate the qualitative evolution of temporal envelope over a set of consecutive time-intervals via bifurcation analysis and control. Bifurcation scenarios involve harmonic asymptotic setsas time-asymptotic limits of harmonic partials. Each harmonic asymptotic set is homeomorphic to a k-tori, where k≤n is the number of contributing harmonic partials. Bifurcation classification of tone colour is then drawn up by encoding the spectral geometry of acoustic signals into harmonic manifolds and harmonic asymptotic sets. For instance, we treat audio C4files obtained from piano and violin. Different bifurcation scenarios distinguish temporal amplitude bifurcations of these sounds. A complete hysteresis cycle is observed within the temporal amplitude bifurcations of C4note played by violin.

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