Transitions of Tonality: A Model-Based Corpus Study

Tonality has been the cornerstone of Western music-theoretical discourse for centuries. This study addresses the subject, using traditional music analysis, data-driven corpus methods, and computational models, concentrating on historical changes of tonality with a particular focus on the 19th century. The thesis engages three analytical levels of increasing scope - micro, meso, and macro - and is thus located between the poles of the particular and the general. The micro-level presents a detailed analysis of Franz Liszt's Sonetto 47 del Petrarca, S. 161, no. 4 (1858), in order to illustrate compositional innovations testifying to the radical changes in tonality within the 19th century. The analysis exemplifies how these novelties permeate musical compositions in that period, and also expose the benefits and limitations of manual music analysis. The meso-level examines a corpus of harmonic annotations of pieces by Beethoven, Schubert, Chopin, Liszt, Dvořák, Grieg, Tchaikovsky, Debussy, and Medtner, containing over 75,000 chord symbols. It presents a comprehensive model for the analysis of chord symbols in large corpora in order to study chords and the progressions between them. Whilst the individual composers' chord vocabularies vary considerably - paying tribute to idiomatic usages of harmony - it is shown that the overarching similarities of the chord distributions point to similarities in their harmonic language that surpass individual traits and that can be modeled by Zipf's and Heaps's laws. An entropy-based method is presented to systematically study the effect of certain features on chord prediction, revealing that suspensions are the strongest predictors. The study shows that chord progressions are largely asymmetrical and proceed mostly by fifths; however, third-based progressions become increasingly prevalent within the studied period. The macro-level explores a corpus of nearly 3 million notes in more than 2000 pieces created by 75 composers, comprising a historical range of approximately 600 years. The encoding of the data distinguishes enharmonically equivalent notes, hence providing a larger note vocabulary than most previous approaches in empirical music research. A Principal Component Analysis (PCA) shows that the line of fifths can be derived from the co-occurrence as well as the co-evolution of tonal pitch-classes. Moreover, the hierarchical topic model known as Latent Dirichlet Allocation (LDA) is used to discover latent tonal profiles. These largely correspond to distributions on contiguous line-of-fifths segments and moreover demonstrate the elevated roles of fifths as well as major and minor thirds as intervals between the most frequent notes. This motivates to model pieces as distributions on the Tonnetz. To that end, a new model, the Tonal Diffusion Model (TDM), is introduced. The results are obtained by fitting the model to the corpus and exhibit two trends. Over the entire historical period under consideration, notes are primarily distributed along the fifths axis of the Tonnetz. Furthermore, 19th-century composers also explore the major and minor thirds axes of the Tonnetz, extending their compositions in ever farther regions. The diverse methodology in this study provides quantitatively grounded insights from a range of perspectives, bridging the fields of music theory, computational musicology, mathematical modeling, and the digital humanities.

Discovering Tonal Profiles with Latent Dirichlet Allocation

Fabian Claude Moss, Martin Alois Rohrmeier

Music analysis, in particular harmonic analysis, is concerned with the way pitches are organized in pieces of music, and a range of empirical applications have been developed, for example, for chord recognition or key finding. Naturally, these approaches rely on some operationalization of the concepts they aim to investigate. In this study, we take a complementary approach and discover latent tonal structures in an unsupervised manner. We use the topic model Latent Dirichlet Allocation and apply it to a large historical corpus of musical pieces from the Western classical tradition. This method conceives topics as distributions of pitch classes without assuming a priori that they correspond to either chords, keys, or other harmonic phenomena. To illustrate the generative process assumed by the model, we create an artificial corpus with arbitrary parameter settings and compare the sampled pieces to real compositions. The results we obtain by applying the topic model to the musical corpus show that the inferred topics have music-theoretically meaningful interpretations. In particular, topics cover contiguous segments on the line of fifths and mostly correspond to diatonic sets. Moreover, tracing the prominence of topics over the course of music history over ∼600 years reflects changes in the ways pitch classes are employed in musical compositions and reveals particularly strong changes at the transition from common-practice to extended tonality in the 19th century.

2021

Mixture models for multivariate extremes

Marc-Olivier Boldi

This thesis is a contribution to multivariate extreme value statistics. The tail of a multivariate distribution function is characterized by its spectral distribution, for which we propose a new semi-parametric model based on mixtures of Dirichlet distributions. To estimate the components of this model, reversible jump Monte Carlo Markov chain and EM algorithms are developed. Their performances are illustrated on real and simulated data, obtained using new representations of the extremal logistic and Dirichlet models. In parallel with the estimation of the spectral distribution, extreme value statistic machinery requires the selection of a threshold in order to classify data as extreme or not. This selection is achieved by a new method based on heuristic arguments. It allows a selection independent of the dimension of the data. Its performance is illustrated on real and simulated data. Primal scientific interests behind a multivariate extreme value analysis reside in the estimation of quantiles of rare events and in the exploration of the dependence structure, for which the estimation of the spectral measure is a means rather than an end. These two issues are addressed. For the first, a Monte Carlo method is developed based on simulation of extremes. It is compared with classical and new methods of the literature. For the second one, an original conditional dependence analysis is proposed, which enlightens various aspects of the dependence structure of the data. Examples using real data sets are given. In the last part, the semi-parametric model and the presented methods are extended to spatial extremes. It is made possible by considering the spectral distribution as the distribution of a random probability, an original viewpoint adopted throughout this thesis. Classical multivariate extremes are extended to extremes of random measures. The application is illustrated on rainfall data in China.

EPFL2004

Statistical characteristics of tonal harmony: A corpus study of Beethoven’s string quartets

Daniel Harasim, Fabian Claude Moss, Markus Franz Josef Neuwirth, Martin Alois Rohrmeier

Tonal harmony is one of the central organization systems of Western music. This article characterizes the statistical foundations of tonal harmony based on the computational analysis of expert annotations in a large corpus. Using resampling methods, this study shows that 1) the rank-frequency distribution of chords resembles a power law, i.e. few chords govern a large proportion of the data; 2) chord transitions are referential and chord predictability is significantly affected by distinguished chord features; 3) tonal harmony conveys directedness in time; and 4) tonal harmony operates differently at the hierarchical levels of chords and keys. These results serve to characterize tonal harmony on empirical grounds and advance the methodological state-of-the-art in digital musicology.

2019