A wavefolder is a very specific design of waveshaper that uses a comparator and some other circuitry. What they do is look to see if the wave goes above (or below) a specific threshold. When it does, instead of clipping off the top and bottom of the wave, they create a mirror image of it and reflect that portion of the wave back upon itself, creating more high harmonics and interesting spectra in the process.
To help drive the incoming waveshape into this behavior, they may have amplifiers on their input, or offsets to cause the wave to clip & fold on one excursion such as positive but not the other (this is sometimes referred to as symmetry). They are often adjustable to allow several folds to occur on a single positive or negative excursion beyond the folding threshold, which causes increasingly bright and noisy sounds on the output.
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