We present the results of a computational investigation of the structure-energy and vibrational properties of alumina under various aggregation states (crystalline, glassy, and liquid) with ab initio procedures. IV-fold, V-fold, and VI-fold oxygen-coordinated aluminum monomeric forms in a dielectric continuum with dielectric constant ε = 4.575 were investigated through DFT/B3LYP gas-phase calculations coupled with a Polarized Continuum Model approach and those of the periodical structure D(6)(3d) (R-3c) which leads to the α-Al(2)O(3) polymorph of alumina, when subjected to symmetry operations, were investigated with the same functional within the LCAO approximation and in the framework of Bloch's theorem. Based on the computed energies and vibrational features, an aggregate of the D(6)(3d) positively charged cluster [Al(12)O(11)](14+) contoured by [AlO(4)](5-) units in an approximate 1:3 proportion to achieve neutrality satisfactorily reproduce the heat capacity of the liquid within experimental uncertainty. The glass is seen as a wrong accretionary form induced by fast cooling rates and subjected to steric forces that locally modify the coordination state of the central atom. Cessation of rotational and translational movements, only partly counterbalanced by acoustic sine-wave-dispersed and excess phonons, gives rise to the huge heat-capacity gap observed at the glass transition (~5.3R). When cooling rates are sufficiently slow, the accretion around the D(6)(3d) seeds follows the structural constraints and the heat capacity of α-alumina is almost perfectly reproduced by the 27 Einstein oscillators coupled with the 3 acoustic terms and the anharmonic corrections.