The purpose of this paper is to develop an analogue of freeness for reduced Cohen-Macaulay spaces embedded in a smooth manifold S which generalizes the notion of Saito free divisors. The main result of this paper is a characterization of freeness in terms of the projective dimension of the module of multi-logarithmic k-forms, where k is the codimension of X in S. We then focus on quasi-homogeneous complete intersection curves, for which we compute explicitly a minimal free resolution of the module of multi-residues and the module of multi-logarithmic forms.