Setting $\rho$ for a desired $T_{60}$

For a desired $T_{60}$, determine the corresponding time constant $\tau$

$$\tau \approx \frac{t_{60}}{6.91}.$$

Use this value in solving for $\rho$,

$$\tau = -\frac{1}{f_1\ln\vert\rho\cos(\pi f_1 T)\vert}$$

$$\ln\vert\rho\cos(\pi f_1 T)\vert = -\frac{1}{f_1\tau}$$

$$\vert\rho\cos(\pi f_1 T)\vert = \displaystyle{e^{-\frac{1}{f_1\tau}}}$$

$$\vert\rho\vert = \frac{e^{-1/(f_1\tau)}}{\vert\cos(\pi f_1 T)\vert}$$