Probabilistic Models of Melodic Interval

Abstract

Two probabilistic models of melodic interval are compared. In the Markov model, the “interval probability” of a note is defined by the corpus frequency of its melodic interval (the interval to the previous note), conditioned on the previous one or two intervals; in the Gaussian model, the interval probability is a simple mathematical function of the size of the note’s melodic interval and its position in relation to the range of the melody. In both models, this interval probability is then multiplied by the probability of the note’s scale degree to yield its actual probability. The two models were tested on four corpora of tonal melodies using cross-entropy. The Markov model yielded a somewhat lower (better) cross-entropy than the Gaussian model, but is also much more complex, requiring far more parameters. The models were also tested on melodic expectation data, and on their ability to predict the distribution of intervals in a corpus. Possible ways of improving the models are discussed, as well as their broader implications for music cognition.