  This paper is a sequel to [8] where we introduced an invariant, called
canonical degree, of Cohen-Macaulay local rings that admit a canonical ideal.
Here to each such ring with a canonical ideal, we attach a different invariant,
called bi-canonical degree, which in dimension 1 appears also in [12] as the
residue of a ring. The minimal values of these functions characterize specific
classes of Cohen-Macaulay rings. We give a uniform presentation of such degrees
and discuss some computational opportunities offered by the bi-canonical
degree.
