Note duration ($T60$)

Note duration: time to decay 60 dB ($T60$), or to 0.001¹ (linear scale), the point of inaudibility.

At $t = T60$, the amplitude is

$$e^{-T60/\tau} = 0.001.$$

Taking the $\log_e$ of both sides yields

$$-T60/\tau = \log_e{(0.001)}.$$

Given a value for $\tau$, the note duration is

$$T60 = -\tau\log(0.001).$$

Given note duration $T60$: the characteristic time constant is

$$\tau = -T60/\log(0.001).$$