Supramolecular assemblies that interact with light have recently garnered much interest as well-defined nanoscale materials for electronic excitation energy collection and transport. However, to control such complex systems it is essential to understand how their various parts interact and whether these interactions result in coherently shared excited states (excitons) or in diffusive energy transport between them. Here, we address this by studying a model system consisting of two concentric cylindrical dye aggregates in a light-harvesting nanotube. Through selective chemistry we are able to unambiguously determine the supramolecular origin of the observed excitonic transitions. These results required the development of a new theoretical model of the supramolecular structure of the assembly. Our results demonstrate that the two cylinders of the nanotube have distinct spectral responses and are best described as two separate, weakly coupled excitonic systems. Understanding such interactions is critical to the control of energy transfer on a molecular scale, a goal in various applications ranging from artificial photosynthesis to molecular electronics.