  This project began as a search for triple-mode pulsating red giants (PRGs)
among known double-mode PRGs,, using AAVSO visual data. It then expanded to
include a check on the reality and significance of some double-mode PRGs. We
identified several "double-mode" PRGs in which one of the periods might simply
be an alias of the other. We also comment on the possibility that the
additional periods might be harmonics, rather than overtones. As for the
triple-mode pulsators: in our sample, we find no convincing cases in which
there is a significant, non-alias third period which might be an additional
radial period. We note, however, that a few triple-mode pulsators have been
found in a very large sample of PRGs in the LMC/ We discuss the results in the
context of recent theoretical models of pulsating red giants.
