Physical temperature and the meaning of the q parameter in Tsallis statistics

We show that the function beta(E) derived from the density of states of a constant heat capacity reservoir coupled to some system of interest is not identical to the physically measurable (transitive) temperature. There are, however, connections between the two quantities as well as with the Tsallis parameter q. We exemplify these connections using the one-dimensional Ising model in the "dynamical ensemble".