We show how two different scales for oscillations between e and mu neu
trinos, characterized by different mixing angles and very different ef
fective mass scales, can arise in a simple and theoretically attractiv
e framework, so as to be relevant to both solar neutrino and accelerat
or-reactor experiments. One scale characterizes direct oscillations, w
hich can accommodate the MSW approach to the solar neutrino problem, w
hereas the other can be considered as arising indirectly, through virt
ual transitions involving the tau neutrino with a mass similar to 1 eV
. This indirect transition allows the possibility of observable <((n)u
)over bar>(mu) <-> <(nu)over bar>(e) oscillations at accelerator and r
eactor energies. We discuss specifically the parameters suggested by a
recent experiment at Los Alamos within this framework.