We compute the covering number by intervals for irrational rotations, that is the largest possible measure of an arbitrarily high Rokhlin stack made with intervals; this gives a topological dynamical invariant which is explicitely linked to the arithmetics of the system through the continued fraction approximation of the argument. Then we compute the covering number by intervals, and give estimations for the measure-theoretic covering number, for exchange of three intervals, which gives new results on their spectral properties.