Obtaining a desired $T_{60}$

The $T_{60}$ is the time to decay to an inaudible level of -60 dB or by 0.001 on a linear scale. Given a loop time of $M$ samples (frequency $f_0$) and a desired $T_{60}$, what should be the value of $g$?

If the loop has a delay of $M$ samples, the number of trips through the loop after $n$ samples, or after $t$ seconds is

$$\frac{n}{M} = \frac{tf_s}{M} = tf_0,$$

where $f_0$ is the fundamental frequency of the loop.

Attenuation at time $t$ is given by

$$\alpha(t) = g^{tf_0}.$$

At time $t=T_{60}$, the attenuation is 0.001,

$$\alpha(T_{60}) = g^{T_{60}f_0} = g^{T_{60}f_s/M} = 0.001,$$

and solving for $g$ yields

$$g = 0.001^{M/(f_sT_{60})}.$$