Intersecting p-branes can be viewed as higher-dimensional interpretations of multicharge extremal p-branes, where some of the individual p-branes undergo diagonal dimensional oxidation, while the others oxidize vertically. Although the naive vertical oxidation of a single p-brane gives a continuum of p-branes, a more natural description arises if one considers a periodic array of p-branes in the higher dimension, implying a dependence on the compactification coordinates. This still reduces to the single lower-dimensional p-brane when viewed at distances large compared with the period. Applying the same logic to the multicharge solutions, we are led to consider more general classes of intersecting p-brane solutions, again depending on the compactification coordinates, which turn out to be described by interacting functions rather than independent harmonic functions. These new solutions also provide a more satisfactory interpretation for the lower-dimensional multicharge p-branes, which otherwise appear to be nothing more than the improbable coincidence of charge-centres of individual constituents with zero binding energy.