A well-established principle of language is that there is a preference for closely related words to be close together in the sentence. This can be expressed as a preference for dependency length minimization (DLM). In this study, we explore quantitatively the degree to which natural languages reflect DLM. We extract the dependencies from natural language text and reorder the words in such a way as to minimize dependency length. Comparing the original text with these optimal linearizations (and also with random linearizations) reveals the degree to which natural language minimizes dependency length. Tests on English data show that English shows a strong effect of DLM, with dependency length much closer to optimal than to random; the optimal English grammar also has many specific features in common with English. In German, too, dependency length is significantly less than random, but the effect is much weaker than in English. We conclude by speculating about some possible reasons for this difference between English and German.

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This study examines the Krumhansl-Schmuckler key-finding model, in which the distribution of pitch classes in a piece is compared with an ideal distribution or "key profile" for each key. Several changes are proposed. First, the formula used for the matching process is somewhat simplified. Second, alternative values are proposed for the key profiles themselves. Third, rather than summing the durations of all events of each pitch class, the revised model divides the piece into short segments and labels each pitch class as present or absent in each segment. Fourth, a mechanism for modulation is proposed; a penalty is imposed for changing key from one segment to the next. An implementation of this model was subjected to two tests. First, the model was tested on the fugue subjects from Bach's Well-Tempered Clavier; the model's performance on this corpus is compared with the performances of other models. Second, the model was tested on a corpus of excerpts from the Kostka and Payne harmony textbook (as analyzed by Kostka). Several problems with the modified algorithm are discussed, concerning the rate of modulation, the role of harmony in key finding, and the role of pitch "spellings." The model is also compared with Huron and Parncutt's exponential decay model. The tests presented here suggest that the key-profile model, with the modifications proposed, can provide a highly successful approach to key finding.