This is the abstract of the Master's Thesis of Balázs Bank.

Abstract
 

The present work is about the synthesis of piano sound based on the grounds of physical principles. For that, first 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 different 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 filters. A new technique is presented to reproduce the attack noise of the piano sound in an efficient and physically meaningful way.
Concerning the implementation issues, a multi-rate piano model is proposed, which resolves the problem of different 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 filter 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 filter as a special case, a novel filter 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 filter. 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 efficient synthesis of other musical instruments as well.

Keywords: digital signal processing, digital waveguide, musical acoustics, musical instruments, piano, sound synthesis