Paul von Hippel & David Huron
Society for Music Perception and Cognition Conference, 1999.
In melodies from a variety of musical cultures, each pitch tends to be close to the pitch before it. This tendency toward pitch proximity has often been construed as an elementary principle of melodic organization. It may be more accurate, however, to view pitch proximity as arising from two more fundamental constraints. The first constraint relates to a melody's "ranginess;" the second constraint relates to a melody's "mobility" within its range. These constraints can be related to simple statistical parameters, leading to a formula for predicting the heights of melodic pitches.
Consideration of these constraints leads to a straightforward explanation of the tendency, long remarked by music theorists, for a melodic skip to be followed by a change of direction. Although this tendency has often been explained in psychological terms, it could plausibly be an artifact of constraints on melodic ranginess. Skips tend toward the extremes of a melody's range; from such extremes, a melody has little choice but to retreat by changing direction. Analyses of melodies from four different continents are generally consistent with this simple explanation for post-skip reversals. After skips, it seems that melodies change direction simply because they lack the space to do otherwise.