  We show that the number of parameters for CM-modules of prescribed rank is
semi-continuous in families of CM rings of Krull dimension 1. This transfers a
result of Knoerrer from the commutative to the not necessarily commutative
case. For this purpose we introduce the notion of ``dense subrings'' which
seems rather technical but, nevertheless, useful. It enables the construction
of ``almost versal'' families of modules for a given algebra and the definition
of the ``number of parameters''. The semi--continuity implies, in particular,
that the set of so-called ``wild algebras'' in any family is a countable union
of closed subsets. A very exciting problem is whether it is actually closed,
hence whether the set of tame algebras is open. However, together with the
results of a former paper of the authors the semi-continuity implies that tame
is indeed an open property for curve singularities (commutative CM rings). An
analogous procedure leads to the semicontinuity of the number of parameters in
other cases, like representations of finite dimensional algebras or finite
dimensional bimodules.
