PolyBLEP

A naive digital sawtooth drops straight from top to bottom. That jump contains harmonics without end, and everything above the Nyquist frequency folds back down as false, inharmonic tones that whistle when you play high up the keyboard. The fix is to replace the ideal jump with a band-limited step, or BLEP. Tim Stilson and Julius O. Smith tackled the problem in 1996, Eli Brandt published minBLEP in 2001, and around 2007 Vesa Välimäki and colleagues showed that a second-order polynomial gets you most of the way.

PolyBLEP corrects only the two samples nearest each discontinuity, using a formula a few lines long. It does not match the heaviest methods, but it costs so little that it has become the default in hobby synths, microcontroller projects and audio-programming coursework.