The recursive Cantor costruction, when applied to planar multilayers under TEM incidence (and multistep transmission lines) leaves both the internal structure of the initiator and the electromagnetic constitutive parameters of the gaps inserted at each recursion step unspecified. The general case where these additional degrees of freedom are included in the design is discussed here, for the first time to the best of our knowledge. A remarkably readable, explicit expression of the reflection coefficient in this general case is obtained under the small-reflections assumption, and briefly discussed.