A system of demand functions is said to exhibit "generalized additive separability" (GAS) if it can be written in the form xi = fi(yi, R(Y)), i = 1,..., n, where yi is the normalized price of the ith good, and R is a function of all normalized prices. That is, GAS implies that "other prices" enter the demand functions only through an index function, R. This paper shows that the demand functions corresponding to directly and indirectly additive utility functions, the Fourgeaud-Nataf demand functions, and Houthakker's self-dual addilog system exhibit GAS. The direct utility functions corresponding to indirect additivity and the self-dual addilog are characterized, and a production function interpretation of some of these results is suggested. "Generalized strong separability" (GSS) and "generalized weak separability" (GWS) are defined, and it is shown that GSS includes both direct and indirect weak separability.