Computer-generated music composition programs have yet to produce creative, natural sounding music. To date, most approaches constrain the search space heuristically while ignoring the inherent structure of music over time. To address this problem, this thesis introduces NEAT Drummer, which evolves a special kind of artificial neural network (ANN) called compositional pattern producing networks (CPPNs) with the NeuroEvolution of Augmenting Topologies (NEAT) method for evolving increasingly complex structures. CPPNs in NEAT Drummer input existing human compositions and output an accompanying drum track. The existing musical parts form a scaffold i.e. support structure, for the drum pattern outputs, thereby exploiting the functional relationship of drums to musical parts (e.g. to lead guitar, bru:is, etc.) The results are convincing drum patterns that follow the contours of the original song, validating a new approach to computergenerated music composition.