  An approximation that is often used in fits to reactor and atmospheric
neutrino data and in some studies of future neutrino oscillation experiments is
to assume one dominant scale, $\Delta m^2$, of neutrino mass squared
differences, in particular, $\Delta m^2_{atm} \sim 3 \times 10^{-3}$ eV$^2$.
Here we investigate the corrections to this approximation arising from the
quantity $\Delta m^2_{sol}$ relevant for solar neutrino oscillations, assuming
the large mixing angle solution. We show that for values of
$\sin^2(2\theta_{13}) \sim 10^{-2}$ (in the range of interest for long-baseline
neutrino oscillation experiments with either intense conventional neutrino
beams such as JHF-SuperK or a possible future neutrino factory) and for $\Delta
m^2_{sol} \sim 10^{-4}$ eV$^2$, the contributions to $\nu_\mu \to\nu_e$
oscillations from both CP-conserving and CP-violating terms involving
$\sin^2(\Delta m^2_{sol}L/(4E))$ can be comparable to the terms involving
$\sin^2(\Delta m^2_{atm}L/(4E))$ retained in the one-$\Delta m^2$
approximation. Accordingly, we emphasize the importance of performing a full
three-flavor, two-$\Delta m^2$ analysis of the data on $\nu_\mu \to \nu_e$
oscillations in a conventional-beam experiment and $\nu_e \to \nu_\mu$,
$\bar\nu_e \to \bar\nu_\mu$ oscillations at a neutrino factory. We also discuss
a generalized analysis method for the KamLAND reactor experiment, and note how
the information from this experiment can be used to facilitate the analysis of
the subsequent data on $\nu_\mu \to \nu_e$ oscillations. Finally, we consider
the analysis of atmospheric neutrino data and present calculations of matter
effects in a three-flavor, two-$\Delta m^2$ framework relevant to this data and
to neutrino factory measurements.
