The objective of this paper is to examine nonlinear normal modes and their bifurcations in cyclic periodic structures. The nonlinear normal modes are computed using a numerical technique that combines shooting and pseudo-arclength continuation. Unlike perturbation techniques, the resulting algorithm can investigate strongly nonlinear regimes of motion. This study reveals that modal interactions may occur without necessarily having commensurate natural frequencies in the underlying linear system. In addition, a countable infinity of such modal interactions are shown to exist in the system.