In recent joint work with Brent Pym, we have proven that the topological K-theory of log Poisson varieties carries a canonical mixed Hodge structure. In this talk I will calculate this mixed Hodge structure in the case of log Poisson tori and show that it is isomorphic to a canonical mixed Hodge structure on the periodic cyclic homology of the corresponding quantum torus. By extracting the 'quantum parameters' from these Hodge structures we will arrive at a nonperturbative calculation of Kontsevich's canonical quantization for toric Poisson structures.