  Following ideas of Connes and Moscovici, we describe two spectral triples
related to the Kronecker foliation, whose generalized Dirac operators are
related to first and second order signature operators. We also consider the
corresponding differential calculi $\Omega_D$, which are drastically different
in the two cases. As a side-remark, we give a description of a known calculus
on the two-dimensional noncommutative torus in terms of generators and
relations.
