In the last decade, many researchers have used modal synthesis for sound generation. Using a modal decomposition, one can convert a large system of coupled differential equations into simple, independent differential equations in one variable. To synthesize sound from the system, one solves these decoupled equations numerically, which is much more efficient than solving the original coupled system. For large systems, such as those obtained from finite-element analysis of a musical instrument, the initial modal decomposition is time-consuming. To design instruments from physical simulation, one would like to be able to compute modes in real-time, so that the geometry, and therefore spectrum, of an instrument can be changed interactively. In this paper, we describe how to quickly compute modes of instruments which have rotational symmetry in order to synthesize sounds of new instruments quickly enough for interactive instrument design.
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