STRUCTURAL COMPLEXITY IN MUSIC

Beginning in the early Renaissance, mathematical transformations of melodic lines became a common musical practice and many composers constructed their pieces according to deliberate structural principles. The prolation canon, whereby a number of simultaneous voices follow the same sequence of notes, each at a different tempo, became a tour de force in the early 15th century and remained popular through the Baroque era. In the following centuries, other forms of complexities, often based on rhythmic shifts or daring pitch variations have been introduced by composers. Various statistical and computational methods have been developed to assess such structural innovations and, in particular, to objectively define the information content and complexity of music. Here we focus on compositions based on discernible melodic subjects or themes, and propose a new, direct measure of structural complexity based on the tonal and rhythmic variations of the defining motifs. For multi-voice compositions with temporal scaling regularities, such as prolation canons, the tonal and rhythmic complexities are reflected in the corresponding fractal indexes. The proposed measures are tested on synthetic examples and also compared with the results of two empirical studies. They are then applied to a number of masterful compositions, single and multi-voice, and are shown to provide discerning insights into musical structures, sensitive to local structural articulations.