Discrete-time modelling of musical instruments

TLDR

Physical modelling techniques for simulating musical instruments—such as mass–spring, modal, wave digital, finite difference, digital waveguide, and source–filter models—are closely related to digital signal processing and are discussed in terms of current trends and future directions. The study aims to discretize musical‑instrument models in time for computer simulation, presenting theory, extensions, and new results on nonlinear and time‑varying digital waveguide models. The authors provide theoretical foundations, discuss extensions, and illustrate each technique with literature examples, such as a vibrating string for digital waveguide and a piano hammer for wave digital filter. The paper presents new results on nonlinear and time‑varying digital waveguide modelling of a string.

Abstract

This article describes physical modelling techniques that can be used for simulating musical instruments. The methods are closely related to digital signal processing. They discretize the system with respect to time, because the aim is to run the simulation using a computer. The physics-based modelling methods can be classified as mass–spring, modal, wave digital, finite difference, digital waveguide and source–filter models. We present the basic theory and a discussion on possible extensions for each modelling technique. For some methods, a simple model example is chosen from the existing literature demonstrating a typical use of the method. For instance, in the case of the digital waveguide modelling technique a vibrating string model is discussed, and in the case of the wave digital filter technique we present a classical piano hammer model. We tackle some nonlinear and time-varying models and include new results on the digital waveguide modelling of a nonlinear string. Current trends and future directions in physical modelling of musical instruments are discussed.