Physical modelling of the piano string scale

Several stages of physics-based mathematical modelling are described for the design of the piano string scale. Strings are assumed to be perfectly flexible, and piano hammers are described by a nonlinear hysteretic model. It is also assumed that the parameters of the hammers for the whole hammer set are determined experimentally beforehand. Simulation procedures are used to systematically adjust the structure of the piano scale to its optimal value. The efficiency of the piano scale is improved by the analysis of the numerically simulated string motion and spectra of the string vibrations excited by the impact of the hammer. The set of variables to be optimized includes the linear mass density and tension of the piano strings and the position of the striking point. In addition, the problem of choosing appropriate tensions for neighbouring strings terminated on separate bass and treble bridges is considered.