A Multidimensional Model for Quantifying Tonal Strength: A Continuous Framework of Analyzing Tonal Evolution Beyond Tonal-Atonal Binary Classification
Authors: Yuliang Li, Nan Nan, Meilian Gu, Xiaohong Guan
Organizations: Frontier Institute of Science and Technology, Xi'an Jiaotong University, Xi'an, China · Interdisciplinary Research Center of Frontier Science and Technology, Xi'an Jiaotong University, Xi'an, China · MOE KLINNS Lab, Faculty of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, China · Department of Music Artificial Intelligence and Music Information Technology, Central Conservatory of Music, China · Center for Intelligent and Networked Systems, Tsinghua University, Beijing, China
Tonality is a fundamental organizing principle of Western music. However, the gradual transition from common-practice tonality to atonality around the turn of the twentieth century cannot be captured by binary labels. To characterize intermediate tonal states and trace this evolution, we propose a multidimensional framework for quantifying tonal strength, grounded in tonal theory and mechanisms underlying tonal dissolution. This framework operationalizes the global statistical regularity, temporal coherence, and local perceptual alignment of tonal organization through three complementary features: pitch-class distribution uniformity, temporal stability of tonal centers, and local tonal clarity, respectively. To validate the model, we introduce two new datasets: a binary dataset comprising 196 tonal and atonal pieces, and a historical MIDI corpus of 1,561 compositions by 21 composers spanning the Baroque era to the early twentieth century. The proposed model achieves high discriminative performance on the binary classification task (F_1=0.985), and quantitatively reveals a continuous historical trajectory from established tonality through progressive weakening to eventual dissolution. Computational analysis of representative composers further confirms significant declines in tonal strength throughout the careers of late-Romantic and modernist figures. Additionally, analysis of EMOPIA dataset reveals a weak but statistically significant positive correlation between higher tonal strength values and positive emotional valence. This work establishes a unified quantitative model for analyzing tonal evolution and provides a continuous measure for future behavioral and neurophysiological investigations of the relationships among tonal organization, musical emotion, and cognition.
How do different musical traditions achieve tonal coherence? Most computational measures to date have analysed tonal coherence in terms of a single dimension, whereas a multi-dimensional analyses have not been sufficiently explored. We propose a new model drawing on the concept of the Tonnetz -- we define two partially independent measures: tonal focus, the concentration of pitch content near a tonal center; and tonal connection, the degree to which pitch content reflects structured intervallic pathways back to that center. Analyzing over 2,800 pieces from Western classical and popular traditions, we find that these traditions occupy overlapping yet distinguishable regions of the two-dimensional space. Popular music shows higher tonal focus, while classical music exhibits higher tonal connection. Our complementary measures ground the differences between different tonal styles in quantitative evidence, and offer interpretable dimensions for computational music analysis and controllable generation.
Weilun Xu, Edward Hall, Martin Rohrmeier
School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, Switzerland · Digital and Cognitive Musicology Lab, École Polytechnique Fédérale de Lausanne, Switzerland
Several tonal pitch spaces and computational models have been proposed to analyze tonal structure in Western tonal music, many of them grounded in principles from music theory and used to support tonal analysis with important implications for tonal tension. In parallel, data-driven methods such as skip-gram have been used to learn chord embeddings from symbolic corpora, but their ability to recover tonal structure and its relation to tonal tension remains underexplored. In this work, we investigate how skip-gram chord embeddings reflect tonal structure and whether they provide a useful basis for analyzing structural aspects of tonal tension. Using chord sequences with and without transposition-based augmentation, we evaluate the learned spaces from geometric, functional, and tension-related perspectives. We show that augmented embeddings exhibit strong transposition equivariance, recover a clear circle-of-fifths structure, and support interpretable shifts between key-related regions of the learned space. We then derive embedding-based measures from chord-to-key distance and contextual chord-distance relations, and show that they capture meaningful aspects of tonal tension structure through correspondence with matched tonal measures and moderate alignment with human tension profiles. Across analyses, transposition-based augmentation generally improves the stability, tonal coherence, and interpretability of the learned space.
Maral Ebrahimzadeh, Gilberto Bernardes, Sebastian Stober
Otto von Guericke University Magdeburg, Artificial Intelligence Lab, Magdeburg, Germany · INESC TEC, Faculty of Engineering, University of Porto, Portugal
We propose a continuous measure of tonal ambiguity that extends the established concept of uniqueness. While uniqueness is widely regarded as necessary for tonality, it cannot (i) discriminate among sets that possess it, (ii) capture hierarchical organization in modes of limited transposition, or (iii) account for temporal unfolding. To address these limitations, we introduce a companion measure, grounded in information theory, that quantifies tonal ambiguity on a continuous scale. The measure applies across pitch-class sets and tuning systems, expanding analytic coverage of tonal relationships and offering a practical tool for theory and analysis.