Waveshaping
Distortion as a pure function: every sample value goes through a curve, and the shape of the curve decides which harmonics appear.
A straight line leaves the signal alone. Bend it gently toward the ceiling — tanh is the classic — and you get soft clipping, the warm saturation that imitates a valve or an overdriven tape machine. A sharp corner gives hard clipping and far uglier harmonics. Marc Le Brun and Daniel Arfib showed in 1979 how Chebyshev polynomials could hit exactly the harmonics you want, turning the technique into a synthesis method rather than just an effect.
The catch is that a nonlinear function creates frequencies well above Nyquist, which fold back into the audible range as inharmonic junk. That is why serious distortion plugins oversample internally, and why cheap ones often sound like a broken radio.