The black-hole–black-string system is known to exhibit critical dimensions and therefore it is interesting to vary the spacetime dimension D, treating it as a parameter of the system. We derive the large D asymptotics of the critical, i.e. marginally stable, string following an earlier numerical analysis. For a background with an arbitrary compactification manifold we give an expression for the critical mass of a corresponding black brane. This expression is completely explicit for Tⁿ, the n-dimensional torus of an arbitrary shape. An indication is given that by employing a higher dimensional torus, rather than a single compact dimension, the total critical dimension above which the nature of the black-brane–black-hole phase transition changes from sudden to smooth could be as low as D ⩽ 11.