Due to spatial and balancing considerations, the implementation of centrifugal pendulum absorbers (CPVA’s) invariably requires that the total absorber inertia be divided into several absorber masses and stationed about the center of rotation. To achieve the designed-for performance, the CPVA's are expected to move in exact unison, since the selection of the total absorber mass is made by assuming an equivalent single absorber mass. In this paper, we determine the conditions under which the unison motion of a system of several identical CPVA’s is dynamically stable. This is done for the special case of tautochronic absorbers subjected to a purely harmonic torque. The stability criterion is obtained by an asymptotic method that exploits certain symmetries in the equations of motion and is based on the assumption that total moment of inertia of the absorbers is much smaller than that of the entire rotating system-an assumption that is almost always satisfied in practice. It is expressed in terms of a critical torque level that is proportional to the square root of the equivalent viscous damping of the individual absorbers. The result is verified by numerical simulations of the system near the critical parameter conditions. A future paper will consider the post-critical response of the system.