Abstract
During the last decade, much attention has been given to sound rendering and the simulation of acoustic phenomena by solving appropriate models described by Hamiltonian partial differential equations. In this contribution, we introduce a procedure to develop appropriate tools inspired from geometric integration in order to simulate musical sounds. Geometric integrators are numerical integrators of excellent quality that are designed exclusively for Hamiltonian ordinary differential equations. The introduced procedure is a combination of two techniques in geometric integration: the semi-discretization method by Celledoni et al. (J Comput Phys 231:6770–6789, 2012) and symplectic partitioned Runge–Kutta methods. This combination turns out to be a right procedure that derives numerical schemes that are effective and suitable for computation of musical sounds. By using this procedure we derive a series of explicit integration algorithms for a simple model describing piano sounds as a representative example for virtual instruments. We demonstrate the advantage of the numerical methods by evaluating a variety of numerical test cases.
Highlights
We introduce a systematic procedure inspired from geometric integration in order to simulate musical sounds
The piano sounds are computed by integrating a Hamiltonian partial differential equation (PDE) model describing the oscillations of the string and an ordinary differential equation (ODE) model describing the dynamics of the hammer
Much attention has been paid to novel approaches to the development of virtual musical instruments, where the PDE models of the components of the instruments
Summary
We introduce a systematic procedure inspired from geometric integration in order to simulate musical sounds In this context, we consider a piano model as a representative example for virtual instruments. Because of these conservation laws, such algorithms often outclass conventional numerical methods in stability and reproducibility of significant phenomena In this regard, the goal of our work is the development of efficient geometric integrators for the models for musical instruments. The procedure introduced in this contribution is a combination of this semi-discretization method and symplectic partitioned Runge–Kutta methods This procedure automatically derives explicit and symplectic integrators for most models for musical instruments. 3 we explain the variational semi-discretization, which is the technique to derive a semi-discrete scheme while preserving the Hamiltonian structure of the equation We apply this approach to the piano model for illustration reasons.
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