  By postulating the relation \theta_{23} \simeq 45^\circ + \eta\theta_{13}, we
seek preferable correction terms to tri-bi-maximal mixing and discuss their
origins. Global analyses of the neutrino oscillation parameters favor \eta=\pm
1/\sqrt{2}; this corresponds to the relation found by Edy, Frampton, and
Matsuzaki some years ago in the context of a T^\prime flavor symmetry. In
contrast, the results of the \nu_\mu disappearance mode reported by the T2K and
Super-Kamiokande collaborations seem to prefer \eta=0, which gives an almost
maximal \theta_{23}. We derive a general condition for ensuring \theta_{23}
\simeq 45^\circ + \eta\theta_{13} and find that the condition is complicated by
the neutrino masses and CP violating phases. We investigate the condition under
simplified environments and arrive at several correction terms to the mass
matrices. It is found that the obtained correction terms can arise from flavor
symmetries or one-loop radiative corrections.
