This research develops a method to determine the set of reachable states for an initially straight, cylindrical triplex tensegrity tower. Supporting cables are actuated to change the overall shape of the structure, with the lengths of struts and face cables held fixed. First, this study establishes the set of geometrically reachable shapes for single-bay cylindrical tensegrities, while considering constraints associated with: collisions between members; and maintaining unilateral member forces (cables in tension and struts in compression). For a single bay, a constrained parameterized transformation describes the physically-reachable range of locations and orientations of the articulated top face with respect to the initial location and orientation of the bottom face. Multi-bay tensegrities enable long, boom-like structures that offer a larger range of obtainable shapes. The articulation of multi-bay tensegrity towers is addressed via sequential application of single-bay transformations. The selection of the number of bays to use in the design of a given tower involves tradeoffs between conventional structural properties (stiffness, strength, mass) and articulation. The results provide insight into the achievable range of motion and potential utility of articulated tensegrity towers.