  We show that the rotation algebras are limit of matrix algebras in a very
strong sense of convergence for algebras with additional Lipschitz structure.
Our results generalize to higher dimensional noncommutative tori and operator
valued coefficients. In contrast to previous results by Rieffel, Li, Kerr, and
Latr\'emoli\`ere we use Lipschitz norms induced by the `carr\'e du champ' of
certain natural dynamical systems, including the heat semigroup.
