The authors investigate the Blandin-Heine-Stroud-Ashcroft suggestion that the Hume-Rothery rules for the phase stability of alpha and beta brasses arise as a consequence of the rapid variation of the free electron density response function in the neighbourhood of 2K_F. Using pseudopotential theory the authors show that the most structure-sensitive contribution to the free energy of a random binary alloy is the one-electron contribution arising from the virtual crystal bandstructure of neutral pseudo-atoms. The calculated difference in the one-electron contribution to the free energy between different crystal structures shows rapid variations at concentrations which are characterised by particular electron per atom ratios Z for which 2K_F is equal to the first or second reciprocal lattice vector of the alloy.