Measuring musics: Notes on modes, motifs, and melodies
Bas Cornelissen

Abstract:
Humans are a musical species: we sing, dance, play, or listen, no matter where we are from. To understand why, musicologists have long studied the rich diversity of musical traditions—or musics—found around the world. One can, for example, compare musics to identify properties that many musical traditions share or properties that very few share. But such questions require you to somehow measure the properties of interest. And that idea motivates this dissertation: can we develop computational methods to measure musics, so that we might compare them? A series of studies, interspersed with lighter interludes, discusses ways to measure modes in plainchant, inventories of melodic and rhythmic motifs, and the shapes of melodies, ending with an intricate rarity: music by Arvo Pärt.

This dissertation primarily analyzes sheet music from a range of musical traditions. In the Catafolk project, we collect a sizeable cross-cultural corpus by bundling several existing corpora, mainly containing German, Chinese, and Native American songs. We also present two corpora of Western plainchant (Cantus Corpus and GregoBase Corpus) and a Python package to parse the plainchant formats. This leads to a series of studies of plainchant. We confirm the melodic arch hypothesis—that phrases tend to be arch-shaped—in plainchant, analyze the predictability of a particular musical connection, and train a small recurrent neural language model to compose new chant artificially.

The centerpiece, however, is a study in which we measure the main organizational structure of plainchant: the eight modes. Modes are melody types that lie somewhere between abstract scales and concrete melodies. We compare three ways to classify musical mode: two approaches that largely view mode as a scale and one distributional approach that focuses on its melodic character. We find that this latter approach can still determine mode fairly accurately even when all pitch information has been discarded. However, this only really works when the mode is segmented in the ‘right’ way: in units corresponding to textual units such as syllables and words.

The smaller units into which music can be decomposed, here called motifs, form the second thread in this dissertation. In the case of plainchant melodies, variable-length motifs corresponding to textual units proved fruitful, but fixed-length motifs can also be helpful when studying rhythmic data. We show how plotting all motifs of three successive temporal intervals in a so-called rhythm triangle effectively highlights rhythmical structures in music and animal vocalizations. It motivates a novel measure of isochronicity—how steady, pulse-like a rhythm is—that generalizes a more commonly used measure (the nPVI). Extending these ideas to melodies, we propose to visualize motifs of three successive notes (or two intervals) in what might be called a melody square to help identify common and rare melodic patterns.

The third thread in this dissertation concerns the shapes of melodies. How can one best represent—measure, if you like—melodic contour? It turns out that one can efficiently describe variability in contour shapes using cosine functions as they closely approximate the principal components of melodies. This leads to a new contour representation, cosine contours, effectively representing the melodic shape using a discrete cosine transform. Cosine contours give a continuous description of contour, while most previous work describes shapes in a discrete fashion, using a fixed set of contour types. We ask if such discrete typologies can accurately describe the variability in contour shape. Rephrasing this as a clustering problem, we propose a way to measure the presence of statistical modes—but find none. This suggests that melodic phrase contours do not cluster into separate types and that discrete typologies may not provide the most appropriate description of melodic contour.

This dissertation ends with a somewhat dissonant finale. Whereas earlier chapters are distant readings of large music collections, the final chapter is a close reading of a single piece: a rarity. Instead of analyzing ‘informal’ music by formal means, we now use formal means to understand the ‘formal’ music of Arvo Pärt. His music is well known to be constructed according to precise mathematical rules, and we attempt to reconstruct the full score of his piece Summa using formal procedures. This formalization makes the constructions that possibly underlie the composition completely transparent. It also highlights the vast range of musical diversity, from a formal composition to a simple folk song. To our understanding that diversity, this dissertation makes only modest contributions. But, if this dissertation inspires new research or new music, its hopes have been fulfilled.