The acoustics of pianos

Abstract

The manufacturing of pianos remains largely empirical, with numerous trial-and-error procedures and fine adjustments at each step of the building process. The “skeleton” of the instrument obeys fundamental principles of vibrations, acoustics, and material science. An abundance of literature is available on its different constitutive parts. However, scientific studies based on a global model of the instrument that connects all of these constitutive parts together are more recent. Such modeling sheds useful light on the essential coupling properties between elements and, in particular, on the string-soundboard coupling at the bridge, and on the radiation of the soundboard. Fine analysis of piano tones also shows that in most cases, a nonlinear model of the strings is necessary to account for perceptually significant features such as precursors in the time-domain and the so-called “phantom partials” in the spectrum. This nonlinearity is based on the coupling between transverse and longitudinal waves in the string. In this lecture, a time-domain model of a complete piano is presented that couples together nonlinear strings, soundboard vibrations, and radiation in air. It highlights the transmission of both transverse and longitudinal string forces to the soundboard, and the influence of rib design and bridge on soundboard mobility and radiation patterns. Comparisons between the results of the model and measurements made on real pianos will be discussed.