We investigate a particular realization of generalized q-differential calculus of exterior forms on a smooth manifold based on the assumption that dN=0 while dk≠0 for k<N. It implies the existence of cyclic commutation relations for the differentials of first order and their generalization for the differentials of higher order. Special attention is paid to the cases N=3 and N=4. A covariant basis of the algebra of such q-grade forms is introduced, and the analogs of torsion and curvature of higher order are considered. We also study a ZN-graded exterior calculus on a generalized Clifford algebra.