A rotational set is a finite subset $A$ of the unit circle $\mathbb{T}=\mathbb{R}/ \mathbb{Z}$ such that the angle-multiplying map $σ_{d}:t\mapsto dt$ maps $A$ onto itself by a cyclic permutation of its elements. Each rotational set has a geometric rotation number $p/q$. These sets were introduced by Lisa Goldberg to study the dynamics of complex polynomial maps. In this paper we provide a necessary and sufficient condition for a set to be $σ_{d}$-rotational with rotation number $p/q$. As applications of our condition, we recover two classical results and enumerate $σ_d$-rotational sets with rotation number $p/q$ that consist of a given number of orbits.