Abstract
There are many ways of synthesizing sound on a computer. The method that we consider, called a mass-spring system, synthesizes sound by simulating the vibrations of a network of interconnected masses, springs, and dampers. Numerical methods are required to approximate the differential equation of a mass-spring system. The standard numerical method used in implementing mass-spring systems for use in sound synthesis is the symplectic Euler method. Implementers and users of mass-spring systems should be aware of the limitations of the numerical methods used; in particular we are interested in the stability and accuracy of the numerical methods used. We present an analysis of the symplectic Euler method that shows the conditions under which the method is stable and the accuracy of the decay rates and frequencies of the sounds produced.
Highlights
Physical sound synthesis uses mathematical models based on the physics of sound production to synthesize sound
The previous literature on mass-spring systems used in sound synthesis describe how these systems work, but have not addressed the issues of the stability and accuracy of the numerical methods
Since the accuracy of the damping depends on both σ and h, we can increase the accuracy of the damping by decreasing the time step h, because, for a fixed value of σ, decreasing h will move the digital damping of the system toward the right side of the figure where the value of σdh is very close to σh
Summary
Physical sound synthesis uses mathematical models based on the physics of sound production to synthesize sound. The previous literature on mass-spring systems used in sound synthesis describe how these systems work (i.e. the equations used and the finite difference equations used to approximate them), but have not addressed the issues of the stability and accuracy of the numerical methods. Using different methods—they do not use the z-transform—they arrive at the stability condition for the undamped mass-spring system as hω0 ≤ 2, which is the same as our results when the damping is zero They do not analyze the symplectic Euler method in terms of frequency warping or its effect on damping. 3. What is the accuracy of the decay rates of the sounds produced by damped mass-spring systems using the symplectic Euler method?
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