Among all positional numeration systems, the widely studied Bertrand numeration systems are defined by a simple criterion in terms of their numeration languages. In 1989, Bertrand-Mathis characterized them via representations in a real base $β$. However, the given condition turns to be not necessary. Hence, the goal of this paper is to provide a correction of Bertrand-Mathis' result. The main difference arises when $β$ is a Parry number, in which case are derived two associated Bertrand numeration systems. Along the way, we define a non-canonical $β$-shift and study its properties analogously to those of the usual canonical one.