Efficient computational modeling of piano strings for real‐time synthesis using mass‐spring chains, coupled finite differences, and digital waveguide sections.

Abstract

In previous waveguide synthesis models for piano, longitudinal waves have been neglected, although it is known that there is audible coupling from transverse to longitudinal vibration in piano strings [e.g., Conklin lectures, http://www.speech.kth.se/music/5/_lectures/]. A general method for accurate 3‐D string simulation is the mass‐spring chain [Rowland and Pask, Am. J. Physics, (1999)], which reduces to half‐coupled wave equations at low amplitude. At yet lower amplitudes, when string slope times string curvature can be neglected, the longitudinal and transverse waves reduce to linear superposition, for which digital waveguides are most efficient for simulation [http://ccrma.stanford.edu//~jos/pasp/]. In this presentation, a hybrid piano‐string model is proposed which employs a mass‐spring chain at the hammer and a digital waveguide model elsewhere. The spatial force vector at either edge of the mass‐spring section drives (and couples) three digital waveguide models corresponding to one longitudinal and two transverse vibrational components. At the bridge, a 3‐D force vector is formed which drives the vertical, horizontal, and longitudinal transfer functions of the soundboard. A simplified 2‐D model neglects one of the transverse planes of vibration. Beyond piano strings, the model can transition adaptively (across time and/or space) among digital waveguides, half‐coupled finite‐difference schemes, and full mass‐spring‐chain models.