Generation of longitudinal vibrations in piano strings: From physics to sound synthesis

Abstract

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.

Model-based sound synthesis of the guqin

“…Initially, P͑z͒ is a full string model, however, as theory and measurements indicate, the phantom partials do not appear at the same frequency range or frequencies as the transverse vibrations. 10,[19][20][21] Consequently, the unwanted low and high frequencies are filtered out with a bandpass filter D͑z͒.…”

“…20 Bank and Sujbert explain this to be a result of transverse to longitudinal coupling which is of a nonlinear nature. 21 Many synthesis models for this phenomenon have been proposed recently. [21][22][23] A computationally efficient solution, that follows the one discussed by Bank and Sujbert, 22 is proposed here.…”

Perception of longitudinal components in piano string vibrations

“…Bank and Sujbert 4 have demonstrated that phantom partials are also generated by the longitudinal motion of the string as a response to the tension variation coming from the transverse string motion. Accordingly, "longitudinal components" refer to both longitudinal modes and phantom partials throughout the paper, and this study seeks the audibility limits of the longitudinal modes and phantom partials together.…”

“…Accordingly, "longitudinal components" refer to both longitudinal modes and phantom partials throughout the paper, and this study seeks the audibility limits of the longitudinal modes and phantom partials together. For a detailed theoretical explanation the reader is referred to the above work 4 or to that of Watzky,5 who has investigated the longitudinal vibration of strings independently.…”

A digital waveguide-based approach for Clavinet modeling and synthesis

“…Some strictly adhere to an underlying physical model and require minimal assumptions, such as finite-difference time-domain methods (FDTD) [10,12]. Modal synthesis techniques, *Correspondence: l.gabrielli@univpm.it 1 Department of Information Engineering, Università Politecnica delle Marche, Via Brecce Bianche 12, Ancona 60131, Italy Full list of author information is available at the end of the article which enable accurate reproduction of inharmonicity and beating characteristics of each partial, have recently become popular in the modeling of stringed instruments [11,[13][14][15]. However, the computational model proposed in this paper is based on digital waveguide (DWG) techniques, which prove to be computationally more efficient than other methods while adequate for reproducing tones of slightly inharmonic stringed instruments [8,16,17] including keyboard instruments [18].…”