We prove that the characteristic Gerstenhaber calculus of a proper log toroidal family together with a log toroidal system of deformations is perfect. This shows that the characteristic Gerstenhaber calculus satisfies the homological assumptions needed to show unobstructedness in the Calabi–Yau case, and thus demonstrates the log toroidal variant of the logarithmic Bogomolov–Tian–Todorov theorem.