A theoretical description of a new experimental technique related to the fifth-order optical nonlinearity of a chromophore in condensed media is presented. Three optical pulses are used to create three consecutive electronic coherence states the duration of the first two of which are controlled. Four nonlinear response functions representative of the full set of 16 response functions are calculated. The wave vectors associated with these four nonlinear response functions are given by k1, 2k3 - k1, 2k3 - 2k2 + k1, and 2k2 - k1. We consider a Gaussian function for the inhomogeneous distribution of electronic transition energies and express the fifth-order three-pulse scattering (FOTS) signals in terms of homogeneous and inhomogeneous contributions. If the two delay times controlling the coherence periods are set equal, the "diagonal" signal appearing with wave vector 2k3 - 2k2 + k1 allows a clean separation of homogeneous and inhomogeneous broadening for a Markovian line broadening function with arbitrary inhomogeneous width. For non-Markovian line broadening functions, the diagonal FOTS signal is free of short-time distortion from the Gaussian components, but in this case, both diagonal and off-diagonal (i.e. unequal delay times) signals must be measured to obtain the homogeneous and inhomogeneous contributions. We illustrate the results in the Markovian limit and provide a preliminary discussion of the situation for non-Markovian line broadening functions. The short-time behavior is discussed and a more general model in which the bath is described as a set of harmonic oscillators characterized by a spectral density is outlined.