Statistical Properties of Tonal Harmony in Bach’s Chorales

This study aims to contribute empirical computational results to the understanding of tonality and harmonic structure. It analyses aspects of tonal harmony and harmonic patterns based on a statistical, computational corpus analysis of Bach’s chorales. This is carried out using a novel heuristic method of segmentation developed specifically for that purpose. Analyses of distributions of single pc sets, chord classes and pc set transitions reveal very different structural patterns in both modes, many, but not all of which accord with standard music theory. In addition, most frequent chord transitions are found to exhibit a large degree of asymmetry, or, directedness, in way that for two pc sets A,B the transition frequencies f(A→B) and f(B→A) may differ to a large extent. Distributions of unigrams and bigrams are found to follow a Zipf distribution, i.e. decay in frequency roughly according to 1/x which implies that the majority of the musical structure is governed by a few frequent elements. The findings provide evidence for an underlying harmonic syntax which results in distinct statistical patterns. A subsequent hierarchical cluster analysis of pc sets based on respective antecedent and consequent patterns finds that this information suffices to group chords into meaningful functional groups solely on intrinsic statistical grounds without regard to pitch