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For many Max/MSP and/or PD users, the [tanh~] object may be familiar. It is likely a much used secret weapon in many contexts! Describing what the tanh does to a signal, both mathematically and sonically can be a headache, and not particularly useful. So I will describe some particularly fun use cases to give a bit of context.
Feedback control:
Feedback is fun. But feedback can get unpredictable.
The user can insert tanh in the feedback path and use it to reign in runaway feedback. Whether it’s feedback in a delay loop, a reverb, filter or straight up microphonic
In this context, it functions as a limiter on feedback amplitude. Instead of fading away to silence, when longer sustain is desired, or running away uncontrollably, a signal can be controlled much more easily with a tanh. There will be a much wider range at which a feedback oscillation will stabilise musically.
Wave shaping:
Running a triangle wave through a tanh will allow for some interesting wave shape control. Fully CCW and the resulting signal will be a slightly attenuated pass through of the original triangle wave. As the level is increased, the corners will soften and the waveform will shape into a sine wave. When a level has increased the sine will soft clip and eventually settle into a unique peaked waveform.
Limiter/distortion:
tanh reacts very well when the user runs a complex audio signal through one of the channels. Drum voices especially! The effect is sonically similar to compression but achieved through wave shaping as opposed to gain reduction. Limiting is a more appropriate description but it definitely has its own unique characteristic. Increasing the level will introduce soft clipping distortion/overdrive.