  We have used systematic, sustained visual observations from the AAVSO
International Database, and the AAVSO time-series analysis package VSTAR to
study the unexplained "long secondary periods" (LSPs) in 27 pulsating red
giants. In our sample, the LSPs range from 479 to 2967 days, and are on average
8.1 +/- 1.3 times the pulsation period. There is no evidence for more than one
LSP in each star. In stars with both the fundamental and first overtone radial
period present, the LSP is more often about 10 times the latter. The visual
amplitudes of the LSPs are typically 0.1 magnitude and do not correlate with
the LSP. The phase curves tend to be sinusoidal, but at least two are sawtooth.
The LSPs are stable, within their errors, over the timespan of our data, which
is typically 25,000 days. The amplitudes, however, vary by up to a factor of
two or more on a timescale of roughly 20-30 LSPs. There is no obvious
difference between the behavior of the carbon (C) stars and the normal oxygen
(M) stars. Previous multicolor photoelectric observations showed that the LSP
color variations are similar to those of the pulsation period, and of the LSPs
in the Magellanic Clouds, and not like those of eclipsing stars. We note that
the LSPs are similar to the estimated rotation periods of the stars, though the
latter have large uncertainties. This suggests that the LSP phenomenon may be a
form of modulated rotational variability.
