We address the interpretation of the splitting between the ground state excitonic transition which indicates the energy of the lowest direct band gap in AlN bulk films and epilayers, and a 36-38 meV higher energy companion. We demonstrate that this splitting is consistent with the initial interpretation in terms of 1s - 2s excitonic splitting by using a calculation of the exciton binding energy which includes mass anisotropy and anisotropy of the dielectric constant. Analytical expressions are proposed to compute the evolution of 1s and 2s excitonic energies using an anisotropy parameter . We show that the values of the dielectric constant that are required to fit the data are  8.7 and  10, values different from the couple of values  7.33 and  8.45 erroneously obtained after a fitting procedure using a spherical description of the long range Coulomb interaction and the classical textbook n-2 spectrum of the excitonic eigenstates. Starting from now, our values are the recommended ones.