Physics-Based Sound Synthesis of the Piano

The present work is about the synthesis of piano sound based on the grounds of physical principles. For that, rst the acoustical properties of the piano have to be understood, since the underlying physical phenomena establish the framework for the model-based sound synthesis. Therefore, the di erent parts of the piano were measured and analyzed. The groundwork of the piano model lies in the digital waveguide modeling of the string behavior. Accordingly, the digital waveguide string model is thoroughly discussed and analyzed. The mathematical equivalence of the digital waveguide and the resonator bank is also presented. The partition of the piano model follows the principles of the sound production mechanism of the real piano. The hammer is modeled by nonlinear interaction. The discontinuity problem arising when connecting the hammer to the string is investigated and new solutions for its avoidance are proposed. The instability problems of the hammer model are overcome by a novel multi-rate implementation. The possible use of a nonlinear damper model is also discussed. The string simulation is based on the digital waveguide. For beating and two-stage decay, a new parallel resonator bank structure is proposed. The soundboard model consists of a feedback delay network with shaping lters. A new technique is presented to reproduce the attack noise of the piano sound in an e cient and physically meaningful way. Concerning the implementation issues, a multi-rate piano model is proposed, which resolves the problem of di erent computational loads presented by the string models of the low and high register. Additionally, the calibration of the piano model is described. A new loss lter design algorithm is presented for the calibration of the digital waveguide. The new technique minimizes the error of the resulting decay times and also ensures the stability of the feedback loop. For the one-pole lter as a special case, a novel lter design technique is proposed. It is founded on the new theoretical results of the Appendix concerning the decay times of a feedback loop containing the one-pole loop lter. A robust technique for the measurement of beating and two-stage decay is presented. This is used for the calibration of the parallel resonator bank. The methods and techniques proposed here are described with the application to piano sound synthesis. Nevertheless, most of them can be exploited for the e cient synthesis of other musical instruments as well.

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