Musical instruments can be classified into two groups---those that have a continuous energy input and produce a sustained sound, and those that have an impulsive energy input and produce a transient decaying sound. Instruments of the first group, which includes violins, flutes, clarinets, trumpets, etc., can be described as ``essentially nonlinear,'' by which it is meant that their musical operation depends critically upon the presence of nonlinearity, and without it they would scarcely sound at all. The transient or percussive group of instruments, which includes pianos, guitars, bells, gongs, and cymbals, on the other hand, can be described as ``incidentally nonlinear'' implying that, while nonlinearity may influence their sound, its presence is by no means essential. Paradoxically, inadequate nonlinearity in a sustained-tone instrument leads to complex behavior, while complexity in a transient instrument is enhanced by strong nonlinearity! This talk explores some of these matters in a simple physical way, with demonstrations from particularly interesting instrument types.