Add to ORKGA 1-factorization M = {M_1,M_2,...,M_n} of a graph G is called perfect if the union of any pair of 1-factors M_i, M_j with i ≠ j is a Hamilton cycle. It is called k-semi-perfect if the union of any pair of 1-factors M_i, M_j with 1 ≤ i ≤ k and k < j ≤ n is a Hamilton cycle. We consider 1-factorizations of the discrete cube Q_d. There is no perfect 1-factorization of Q_d, but it was previously shown that there is a 1-semi-perfect 1-factorization of Q_d for all d. Our main result is to prove that there is a k-semi-perfect 1-factorization of Q_d for all k and all d, except for one possible exception when k=3 and d=6. This is, in some sense, best possible. We conclude with some questions concerning other generalisations of perfect 1-factorizati...