topological modeling of tone-color in music: tone-color limit sets and cw complexes

We propose a topological approach for timbre (tone-color) modeling of musical sounds using tone-color limit sets and cw complex decompositions. we explain how we hear a cone on a torus and how it is related to the timber of a musical sound. a hearing threshold reduction will cause the partial collapses of these toral cones. in other words, any reduction inyour hearing ability will thus cost you to instead hear cones on lower dimensional tori. we here pay less attention to the convergent speed evolution of audio signals and deal with acoustic signals by observing them as of their phase portraits. in this direction, we introduce a parametric space with a cw structure, where it describes the relative subsound normalizedintensity variations and accommodates their qualitative evolutions. we propose a toral manifoldfor modeling the leading harmonic partials of complex musical sounds as per the spectral geo-metric modeling of their tone-color. the signal of a complex sound would then live in a compactsub-manifold called by tone-color manifold of the complex sound. tone-color manifold is associ-ated with a parameter from the parametric space equipped with the cw structure. tone-colormanifold here is homeomorphic either to a toral cylinder or a cone on a torus. the toral dimen-sion n of the tone-color manifold is associated with the number of subsounds. subsounds of acomplex sound always converge either to an attracting n-torus or to the origin. the attractingtori are called by the time-forward tone-color limit set of the sound.