  Recently Davoudiasl {\it et al} \cite{steinhardt} have introduced a new type
of interaction between the Ricci scalar $R$ and the baryon current $J^{\mu}$,
${\partial_\mu R} J^{\mu}$ and proposed a mechanism for baryogenesis, the
gravitational baryogenesis. Generally, however, $\partial_{\mu} R$ vanishes in
the radiation dominated era. In this paper we consider a generalized form of
their interaction, $\partial_{\mu}f(R)J^{\mu}$ and study again the possibility
of gravitational baryo(lepto)genesis. Taking $f(R)\sim \ln R$, we will show
that $\partial_{\mu}f(R)\sim \partial_{\mu} R/R$ does not vanish and the
required baryon number asymmetry can be {\it naturally} generated in the early
universe.
