  To any continuous eigenvalue of a Cantor minimal system $(X,\,T)$, we
associate an element of the dimension group $K^0(X,\,T)$ associated to
$(X,\,T)$. We introduce and study the concept of irrational miscibility of a
dimension group. The main property of these dimension groups is the absence of
irrational values in the additive group of continuous spectrum of their
realizations by Cantor minimal systems. The strong orbit equivalence
(respectively orbit equivalence) class of a Cantor minimal system associated to
an irrationally miscible dimension group $(G,\,u)$ (resp. with trivial
infinitesimal subgroup) with trivial rational subgroup, have no non-trivial
continuous eigenvalues.
