Large-unit-cell complex metallic alloys (CMAs) frequently achieve stability by lowering the kinetic energy of the electron system through formation of a pseudogap in the electronic density of states (DOS) across the Fermi energy εF. By employing experimental techniques that are sensitive to the electronic DOS in the vicinity of εF, we have studied the stabilization mechanism of two binary CMA phases from the Al-Mg system: the γ-Mg17Al12 phase with 58 atoms in the unit cell and the β-Mg2Al3 phase with 1178 atoms in the unit cell. Since the investigated alloys are free from transition metal elements, orbital hybridization effects must be small and we were able to test whether the alloys obey the Hume-Rothery stabilization mechanism, where a pseudogap in the DOS is produced by the Fermi surface-Brillouin zone interactions. The results have shown that the DOS of the γ-Mg17Al12 phase exhibits a pronounced pseudogap centered almost exactly at εF, which is compatible with the theoretical prediction that this phase is stabilized by the Hume-Rothery mechanism. The disordered cubic β-Mg2Al3 phase is most likely entropically stabilized at high temperatures, whereas at lower temperatures stability is achieved by undergoing a structural phase transition to more ordered rhombohedral β' phase at 214 ° C, where all atomic sites become fully occupied. No pseudogap in the vicinity of εF was detected for the β' phase on the energy scale of a few 100 meV as determined by the 'thermal observation window' of the Fermi-Dirac function, so that the Hume-Rothery stabilization mechanism is not confirmed for this compound. However, the existence of a much broader shallow pseudogap due to several critical reciprocal lattice vectors [Formula: see text] that simultaneously satisfy the Hume-Rothery interference condition remains the most plausible stabilization mechanism of this phase. At Tc = 0.85 K, the β' phase undergoes a superconducting transition, which slightly increases the cohesive energy and may contribute to relative stability of this phase against competing neighboring phases.