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Paper IPM / M / 17116  


Abstract:  
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 nleading harmonic partials and an amplitude spectral nvector A for n â¥2. We introduce harmonic manifolds associated with amplitude spectral vectors. This is to accommodate the nleading harmonic partials of the note. Then, harmonic partials are modelled by solutions of a radial coupling of ntuple parametric Hopf oscillators. We use one bifurcation parameter and interpolate the qualitative evolution of temporal envelope over a set of consecutive timeintervals via bifurcation analysis and control. Bifurcation scenarios involve harmonic asymptotic setsas timeasymptotic limits of harmonic partials. Each harmonic asymptotic set is homeomorphic to a ktori, where kâ¤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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