The preceding discussion considered several possible approximations for nonlinear piano-string synthesis. Other neglected terms in the stiff-string wave equation were not even discussed, such as terms due to shear deformation and rotary inertia that are included in the (highly accurate) Timoshenko beam theory formulation [261,169]. The following questions naturally arise:
- How do we know for sure our approximations are inaudible?
- We can listen, but could we miss an audible effect?
- Could a difference become audible after more listening?
Note that there are software tools (e.g., from the world of perceptual audio coding [62,472]) that can be used to measure the audible equivalence of two sounds . For an audio coder, these tools predict the audibility of the difference between original and compressed sounds. For sound synthesis applications, we want to compare our ``exact'' and ``computationally efficient'' synthesis models.
A Stiff Mass-Spring String Model
Efficient Waveguide Synthesis of Nonlinear Piano Strings