audio - Precise Centre frequency of an All-pole digital filter

I'm not an engineer and have essentially taught myself all I know - this present problem is giving me some problems.

I have an all-pole filter (a gammatone filter - impulse response is essentially a damped cosine) implemented as a cascade of identical 2nd-order pole pair sections. The poles are located in the z-domain at $\exp(-\beta + i\theta)$: $\beta$ is the bandwidth in radians/sample and $\theta$ is the 'ringing' frequency in radians/sample.

Given a value for $\theta$, this pole pair does not produce a filter with peak gain at the frequency $\theta$ - I'm trying to find out exactly what the offset is (i.e. at what frequency a filter with a given $\theta$ actually has peak magnitude), so that I can produce filters that peak at precisely a given arbitrary frequency.

I've found an equation for calculating what this offset is given poles located in the s-domain at $-\beta + i\theta$:

$$\mathrm{f_{c,actual}} = \theta \sqrt{1 - \dfrac{\beta^2}{\theta^2}}$$

This formula provides results consistent with observations up to a certain frequency, though I believe it needs to be modified to produce results that work for arbitrary discrete spectrums.. I've tried a number of things and just can't quite figure it out.

Any help is greatly appreciated.