We present a formula approximating the mean escape time (MST) of a particle from a tilted multi-periodic potential well. The potential function consists of a weighted sum of a finite number of component functions, each of which is periodic. For this particular case, the least period of the potential function is a common period amongst all of its component functions. An approximation of the MST for the potential function is derived, and this approximation takes the form of a product of the MSTs for each of the individual periodic component functions. Our first example illustrates the computational advantages of using the approximation for model validation and parameter tuning in the context of the biological application of DNA transcription. We also use this formula to approximate the MST for an arbitrary tilted periodic potential by the product of MSTs of a finite number of its Fourier modes. Two examples using truncated Fourier series are presented and analyzed.