For the gauge group ${[\mathrm{SU}(6)]}^{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ which unifies nongravitational forces with flavors we analyze the generational seesaw mechanism. At the tree level we get ${m}_{{\ensuremath{\nu}}_{e}}=0$ and ${m}_{{\ensuremath{\nu}}_{\ensuremath{\tau}}}\ensuremath{\sim}{m}_{{\ensuremath{\nu}}_{\ensuremath{\mu}}}\ensuremath{\sim}\frac{{M}_{L}^{2}}{{M}_{H}}$, where ${M}_{L}\ensuremath{\sim}{10}^{2}$ GeV and ${M}_{H}\ensuremath{\ge}{10}^{5}$ GeV are the weak and horizontal interactions mass scales, respectively. The right-handed neutrinos get Majorana masses ${M}_{R}$ of the order of the scale where ${\mathrm{SU}(2)}_{R}$ is broken. A low energy exotic neutral lepton with a mass of the order of $\frac{{M}_{H}^{2}}{{M}_{R}}$ is predicted. Radiative corrections can produce ${m}_{{\ensuremath{\nu}}_{e}}\ensuremath{\ne}0$, four orders of magnitude smaller than the other neutrino masses.