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.
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