Tone sandhi is a common occurrence in different varieties, the most famous being Mandarin Chinese, in which a third tone (high-low-high, or falling-rising) followed by another third tone becomes a low-high or rising tone. Traditional accounts of tonal assimilation are argued against, in that they fail to account for the specific outcome of the changed tone, and especially fail when applied to other Chinese dialects with much more complex tone sandhi phenomena. 
    Historical and cross-dialectal data are presented, showing that these tonal changes seem to preserve archaic tone values and features. The regular, unchanged tonal values represent the preferred default underlying representations, while the sandhi (changed tone) values occur only in specific contexts, namely, in certain tone combinations. These facts, and the wholly arbitrary values of the sandhi tones, indicate that they are historical relics and operate in the grammars of modern dialects as morphophonemic material. Their appearance in restricted contexts can be accounted for by directional alignment constraints that override other faithfulness constraints, along with constraints that require faithful outputs for prosodic heads. A set of such constraints is proposed to account for Mandarin sandhi. The same kinds of constraints also work in a non-cyclical form of optimality theory which uses output-output constraints for more complex forms of sandhi, as shown for Tianjin dialect.