  In the extension of the standard model with one right-handed neutrino and one
Higgs triplet, we propose a suppression mechanism, obtaining small masses for
the active neutrinos, while mixing angles are predicted with a right-handed
neutrino at the TeV scale and Yukawa couplings at the order of
$\mathcal{O}(1)$. In this extension, the seesaw formula is proportional to the
difference between two Yukawa couplings: the one that governs the interactions
of the ordinary matter through the Higgs triplet, and the coupling of the new
neutrino through the scalar doublet, so that by aligning both Yukawa couplings,
exact zero-mass active neutrinos are obtained. By perturbating this alignment
condition, we obtain neutrino masses proportional to the magnitude and
direction of the perturbation in the flavour space. Bimaximal and nearly
bimaximal mass structures emerge from specific unalignment forms.
