This paper describes an eigenvalue inclusion principle for a simple, rotationally periodic structure P whose i-th substructure Si is connected to a neighboring substructure Si+1 through a single-degree-of-freedom interface constraint Ii+1. The state vector vi+1 at the interface Ii+1, consisting of the displacement and the force at the interface, is represented in terms of the state vector vi at the interface Ii through transfer functions of the substructure Si. The periodicity of the structure P then requires that a linear combination of the transfer functions of Si be zero. As a consequence, a simple periodic structure P with period N will have exactly N eigenvalues lying between two consecutive eigenvalues of the substructure Si. Finally, this eigenvalue inclusion property is illustrated on a periodic structure with known exact eigensolutions.