Wavefolding

Subtractive synthesis starts with a waveform that has many harmonics and removes some of them with a filter. Wavefolding starts with a waveform that has few harmonics, usually a sine or a triangle, and adds more by changing its shape. A wavefolder is a kind of waveshaper. Wherever the input goes past the folder's threshold, the output turns back towards the centre, as if the waveform had been folded over at that level. A sine wave driven just past the threshold gains a small dip at each peak. Driven further, the dip goes past the opposite threshold and folds again, so each cycle contains more turns and the output contains more harmonics, reaching higher up the spectrum.

The tool below folds a sine wave. Drive sets how far past the threshold it goes, symmetry adds an offset so that the two halves of the cycle fold differently, and the plot shows one cycle of the output and the levels of its harmonics.

Drive, folds and harmonics

Each time the driven waveform passes another threshold, a new fold appears in the cycle. More turns per cycle means faster changes in the waveform, and faster changes mean higher harmonics, so the spectrum grows upwards as the drive rises. The levels of individual harmonics do not simply rise, though. Some grow, fall back and grow again as the folds move through the cycle, which gives a folder sweep its shifting, vocal quality. For the smooth fold with the symmetry at zero, the harmonic levels follow the same Bessel functions that set the sideband levels in FM, with an index proportional to the drive. A sharp folder, whose turns are corners, adds more high harmonics than a smooth one, whose turns are rounded.

Symmetry

A sine wave folded evenly about its centre keeps the same shape in the second half of each cycle as in the first, upside down, so like a square wave it contains only odd harmonics. Adding an offset before the folder moves the wave towards one threshold, so the positive and negative halves cross the thresholds at different points and can fold different numbers of times. The two halves are no longer mirror images, and even harmonics appear.

West coast

Don Buchla, who built synthesisers in California from the 1960s, made wavefolding central to his instruments, notably in the timbre section of the Buchla 259 complex waveform generator. The approach of adding harmonics to a simple waveform by shaping it, instead of filtering a waveform that already has many, is often called west coast synthesis. It is contrasted with the east coast, subtractive approach of the instruments built in New York State by the engineer Robert Moog. In practice the two are often combined, and a wavefolder followed by a lowpass filter is a common modern voice.