  Pioneering atmospheric muon neutrino experiments have demonstrated the
near-maximal magnitude of the flavor mixing angle $\theta_{23}$. But the
precise value of the deviation $D \equiv 1/2 - \sin^2 \theta_{23}$ from
maximality (if nonzero) needs to be known, being of great interest --
especially to builders of neutrino mass and mixing models. We quantitatively
investigate in a three generation framework the feasibility of determining $D$
in a statistically significant manner from studies of the atmospheric
$\nu_\mu,\bar\nu_\mu$ survival probability including both vacuum oscillations
and matter effects. We show how this determination will be sharpened by
considering the up-down ratios of observed $\nu_\mu$- and $\bar\nu_\mu$-induced
events and the differences of these ratios in specified energy and zenith angle
bins. We consider 1 Megaton year of exposure to a magnetized iron calorimeter
such as the proposed INO detector ICAL, taking into account both energy and
zenith angle resolution functions. The sensitivity of such an exposure and the
dependence of the determination of $D$ on the concerned oscillation parameters
are discussed in detail. The vital use of matter effects in fixing the octant
of $\theta_{23}$ is highlighted.
