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The manipulation of suspended particles using ultrasonic fields remains an active area of research due to the many potential applications of acoustophoresis. While prior research has focused primarily on the acoustophoretic behavior of particles in response to an acoustic field of a single wavelength, relatively little work has been done to study the acoustophoretic behavior of subject to two or more concurrent frequencies of sound. The presence of multiple concurrent, resonant frequencies can lead to more complicated particle behavior spatially across the chamber than the simple node/anti-node attraction/repulsion behavior present in single-frequency chambers. In this study, we investigate the behavior of spherical particles (both single phase and core-shell) in multi-frequency chambers in both the Rayleigh (long wavelength) limit and cases where the particle size is the same order of magnitude as the wavelength of the background field. A mathematical description of the phenomena is developed in terms of the ratios of the pressure amplitudes for the individual background frequencies, including the criteria for continuously moveable nodes. Attention will also be paid to the implications of more than one simultaneous frequency on the motion of particles undergoing a change in size.