  We investigate the generation of small neutrino masses in a clockwork
framework which includes Dirac mass terms as well as Majorana mass terms for
the new fermions. We derive analytic formulas for the masses of the new
particles and for their Yukawa couplings to the lepton doublets, in the
scenario where the clockwork parameters are universal. When the Majorana masses
all vanish, the zero mode of the clockwork sector forms a Dirac pair with the
active neutrino, with a mass which is in agreement with oscillations
experiments for a sufficiently large number of clockwork gears. On the other
hand, when the Majorana masses do not vanish, neutrino masses are generated via
the seesaw mechanism. In this case, and due to the fact that the effective
Yukawa couplings of the higher modes can be sizable, neutrino masses can only
be suppressed by postulating a large Majorana mass for all the gears. Finally,
we discuss the constraints on the mass scale of the clockwork fermions from the
non-observation of the rare leptonic decay $\mu\rightarrow e\gamma$.
