Dedekind different and type sequences

Let R be a one-dimensional, local, Noetherian domain. We assume R analitycally irreducible and residually rational. Let ω be a canonical module of R such that R ω R and let θ_D := R : ω be the Dedekind different of R. Our purpose is to study how θ_D is involved in the type sequence of R and to compare the type sequence of R with the type sequence of θ_D (for the notion of type sequence we refer to [11], [1] and [13]). These relations yield some interesting consequences.