The notion of a particular kind of ordering (Markovian) in one-dimensional alloys is introduced. It is shown how, within the context of the tight-binding model, one may obtain a hierarchy of ever better approximations for the electronic density of states of a Markhoff alloy. Some detailed calculations for the case of 1:1 binary alloys with varying degrees of diagonal and off-diagonal disorder are presented. It is found that the second approximation of the hierarchy already gives very fair agreement with known exact results for the density of states. Generalizations of the method to calculate near-neighbor Green's functions and to apply to alloys with more elaborate (generalized Markovian) orderings are disucssed. © 1979 The American Physical Society.