  In the derivation of the Standard Model from the axioms of Noncommutative
Geometry, the scalar sector is given by a finite Dirac operator which has to
satisfy the so-called \emph{first-order condition}. However, the general
solution to this constraint still has unphysical terms which must be fine-tuned
to zero. Moreover, the first-order condition generally does not survive in
extensions to models with gauge groups larger that $U(1)\times SU(2)\times
SU(3)$. In this paper we show that in the $U(1)_{\rm B-L}$-extension one can
implement a weaker form of the first-order condition which we argue is
necessary in order for Noncommutative Gauge Theory to make sense at all, and
that this condition reduce the amount of fine-tuning to the off-diagonal terms
in the Yukawa mass matrices for the leptons and quarks. We also show that this
condition eliminates the Majorana mass terms for right-handed neutrinos when it
is applied to the Pati-Salam model.
