One approach to the generation of natural-sounding synthesized speech waveforms is to select and concatenate units from a large speech database. Units (in the current work, phonemes) are selected to produce a natural realisation of a target phoneme sequence predicted from text which is annotated with prosodic and phonetic context information. We propose that the units in a synthesis database can be considered as a state transition network in which the state occupancy cost is the distance between a database unit and a target, and the transition cost is an estimate of the quality of concatenation of two consecutive units. This framework has many similarities to HMM-based speech recognition. A pruned Viterbi search is used to select the best units for synthesis from the database. This approach to waveform synthesis permits training from natural speech: two methods for training from speech are presented which provide weights which produce more natural speech than can be obtained by hand-tuning.