Envelopes: Exponential

For percussive, plucked or struck instrument sounds, the envelope needs to model an exponential decay. This is very useful for string-like sounds but most importantly for most electronic musicians, it is the very core of kick drum sounds.

In contrast to the ADSR envelope, the exponential one does not contain a sustain portion for holding a sound. The only parameter is the decay rate, allowing quick adjustment. Alternative to an actual exponential, a modified reciprocal function can be used for easier implementation. The factor $d$ controls the rate of the decay, respectively the decay time:

$$ e = \frac{1}{(1+(d t))} $$

The following example adds a short linear attack before the exponential decay. This minimizes clicks which otherwise occur through the rapid step from $0$ to $1$:

Attack Time:

Decay Time: