  We show how two different scales for oscillations between $e$ and $\mu$
neutrinos, characterized by different mixing angles and effective mass scales,
can arise in a simple and theoretically attractive framework. One scale
characterizes direct oscillations, which can accommodate the MSW approach to
the solar neutrino problem, whereas the other can be considered as arising
indirectly, through virtual transitions involving the $\tau$ neutrino with a
mass $\sim$ 1 eV. This indirect transition allows the possibility of observable
$\bar \nu_\mu \leftrightarrow \bar \nu_e$ oscillations at accelerator and
reactor energies. We discuss specifically the parameters suggested by a recent
experiment at Los Alamos within this framework.
