Starting with combinatorial degeneration data, Gross and Siebert found a canonical formal smoothing of the associated degenerate Calabi-Yau space. In joint work with Siebert, I compute certain period integrals for these families and show that the mirror map is trivial. In other words, the canonical coordinate of Gross-Siebert is a canonical coordinate in the sense of Hodge theory as defined by Morrison. As a consequence, the formal families lift to analytic families. We introduce tropical 1-cycles that we turn in to ordinary n-cycles whose periods have a simple log pole. We show that such cycles generate the dual of the tangent space to the CY moduli space.