Longitudinal vibration of piano strings greatly contributes to the distinctive character of low piano notes. In this paper a simplified modal model is developed, which describes the generation of phantom partials and longitudinal free modes jointly. The model is based on the simplification that the coupling from the transverse vibration to the longitudinal polarization is unidirectional. The modal formulation makes it possible to predict the prominent components of longitudinal vibration as a function of transverse modal frequencies. This provides a qualitative insight into the generation of longitudinal vibration, while the model is still capable of explaining the empirical results of earlier works. The semi-quantitative agreement with measurement results implies that the main source of phantom partials is the transverse to longitudinal coupling, while the string termination and the longitudinal to transverse coupling have only small influence. The results suggest that the longitudinal component of the tone can be treated as a quasi-harmonic spectrum with formantlike peaks at the longitudinal modal frequencies. The model is further simplified and applied for the real-time synthesis of piano sound with convincing sonic results.

This paper reviews recent developments in physics-based synthesis of piano. The paper considers the main components of the instrument, that is, the hammer, the string, and the soundboard. Modeling techniques are discussed for each of these elements, together with implementation strategies. Attention is focused on numerical issues, and each implementation technique is described in light of its efficiency and accuracy properties. As the structured audio coding approach is gaining popularity, the authors argue that the physical modeling approach will have relevant applications in the field of multimedia communication.

This paper presents a real-time piano synthesizer where both the transverse and longitudinal motion of the string is modeled by modal synthesis, resulting in a coherent and highly parallel model structure. The paper applies recent developments in piano modeling and focuses on the issues related to practical implementation (e.g., numerical stability, aliasing, and efficiency). A strong emphasis is given to modeling nonlinear string vibrations, and a new variation of earlier synthesis techniques is proposed which is particularly well suited for modal synthesis. For soundboard modeling, the possibilities of using fast Fourier transform-based fast convolution and parallel second-order filters are discussed. Additionally, the paper describes the details of the software implementation and discusses the computational complexity of each model block. The piano model runs on current computer hardware with full polyphony in real time

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