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Numerous infrared spectroscopic observations were obtained of nine asymptotic giant branch (AGB) field M giants that have multiple periods of light variability. Each star has a short period of several months, which is typical of low amplitude pulsation for stars on the AGB, as well as a long period of 1–3 yr, which is significantly longer than the predicted fundamental-mode pulsations for these stars. The location of these stars in the AGB period-luminosity relation is discussed. For six of the nine giants we found radial-velocity periods that confirm the long-period light variability. Although we considered the possibility that the velocity variations result from orbital motion, we conclude that the long-period velocity changes in most, if not all of the sample stars, likely result from a currently unknown type of pulsation rather than duplicity.