Modal synthesis is simple in theory: a method of synthesizing sound where an excitation signal is fed into a parallel bank of resonators [Bilbao2006]. Each resonator, known as a "mode," has the impulse response of an exponentially decaying sine wave and has three parameters: frequency, amplitude, and decay time. It's a powerful method that excels in particular for pitched percussion like bells and mallet instruments.
Despite its conceptual simplicity, modal synthesis has a daunting number of parameters to tune -- three times the number of modes, to be exact -- and setting them by hand isn't practical or musically expressive. How are we supposed to get great sound design when we have so many degrees of freedom? In interface design parlance, what we want is a divergent mapping [Rovan1997] that takes a small number of controls, maybe four knobs, and maps them to the several dozen frequencies, amplitudes, and decay times that control the modal synthesizer.
While it is possible to reverse engineer modes from a recorded sample (see [Ren2012] for a particularly impressive application), I'm most interested in "parametric" modal synthesis where each model derives from mathematical formulas that allow the user to navigate around a multidimensional space of timbres. I found a number of such algorithms while shopping around the literature from both computer music and mechanical engineering, so I decided to summarize them in a concise reference for sound designers and composers.