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Emergent Tonal Structure in Learned Chord Embeddings and Its Relation to Tonal Tension
Maral Ebrahimzadeh, Gilberto Bernardes, Sebastian Stober
TL;DR
Whether learned chord embeddings capture tonal, transpositional, and tension-related structure remains an open question. The paper evaluates skip-gram embeddings trained with and without transposition augmentation and derives embedding-based tension measures. Augmentation generally strengthens tonal organization, while the tension analysis captures a meaningful subset of structure and shows moderate agreement with human ratings.
Problem
Whether simple chord-level embeddings capture richer tonal, transpositional, and tension-related structure beyond local harmonic similarity remains underexplored.
Source: §1 · p. 1Method
The study trains skip-gram chord embeddings on original and transposition-augmented sequences and analyzes chord-to-key distances and contextual chord-distance relations as tension measures.
Source: §1 · p. 1, §5.1 · p. 4, §5.3 · p. 5Results
Transposition augmentation generally strengthens the embeddings’ tonal organization, while embedding distances correspond to matched TIV quantities and a dynamic measure shows moderate agreement with human ratings.
Source: §6 · p. 6Takeaways & Limitations
The learned spaces capture a meaningful subset of tonal tension, making the analysis diagnostic rather than a complete perceptual model.
Source: §6 · p. 6Because chord labels omit inversion, realized bass motion, voicing, rhythmic salience, and non-chord tones, the human-rating experiment is diagnostic rather than a full predictor.
Source: §5.3 · p. 6
Abstract
from arXiv · showhide
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.
1. INTRODUCTION
Tonal spaces offer interpretable accounts of relationships among notes, chords, and keys, and support analyses including tonal tension. Whether learned chord embeddings capture richer tonal, transpositional, and tension-related structure remains underexplored; this study examines skip-gram embeddings trained with and without transposition augmentation.
1. INTRODUCTION
- Tonal pitch spaces provide interpretable descriptions of tonal structure and support tonal similarity, key finding, and analyses of tonal stability, expectation, and tension.Computational accounts often model perceived tension through distances and movements within a tonal space.Source: §1 · p. 1
- Skip-gram chord embeddings learn from co-occurrence patterns alone, and prior work found functional chord relationships, key relations, and implicit circle-of-fifths structure in slice-based representations.This suggests tonal structure can emerge from distributional learning in relatively simple embedding spaces.Source: §1 · p. 1
- Whether simple chord-level embeddings capture richer tonal, transpositional, and tension-related structure beyond local harmonic similarity remains underexplored.The question matters in a field where tonal spaces are important to music theory, cognition, and tonal-tension research.Source: §1 · p. 1
- The study evaluates skip-gram chord embeddings trained on original and transposition-augmented sequences through geometric, functional, and tension-related analyses.Code is publicly available.Source: §1 · p. 1
2. RELATED WORK
Prior tonal-tension models commonly rely on hand-designed tonal geometries, while learned representations have also shown tonal structure across architectures and input granularities. This work uses chord-token embeddings to examine transposition behavior and their support for key-directed and tension-related analyses.
2. RELATED WORK
- Hand-designed tonal geometries, sometimes enriched with music-theoretic priors, have commonly provided the basis for computational tension analysis.Examples include spiral-array geometry and Tonal Pitch Space, which have supported computational models of tension.Source: §2 · pp. 1–2
- Learned chord and pitch-class representations have been used in downstream tasks, while unsupervised latent spaces have also shown circle-of-fifths geometry and key-related structure.Together, these studies suggest tonal structure can emerge across architectures and input granularities.Source: §2 · p. 2
- Compared with Chuan et al.’s polyphonic-slice embeddings, this work uses chord-token embeddings aligned more directly with chord-based tonal analysis.It also examines transposition effects, key-directed steering, and tension-related probes.Source: §2 · p. 2
3. DATA AND TRAINING SETUP
The study trains skip-gram embeddings on two annotated harmony datasets, reducing supported chord labels to major and minor triads. It compares original and transposition-augmented training settings using a fixed model setup.
3.1 Datasets
- BPS-FH supplies expert Roman-numeral annotations for Beethoven sonata first movements, while Isophonics supplies chord annotations for the Beatles catalogue.Only the chord-label layer of Isophonics is used; chord roots and qualities are derived from BPS-FH annotations.Source: §3.1 · p. 2
- Supported labels are reduced to 24 major- and minor-triad tokens while preserving chord roots; sevenths and inversions are not retained, and voicing is not encoded.Major-, minor-, and dominant-family labels, including selected extensions, map to corresponding triads.Source: §3.1 · p. 2
- After filtering and triad reduction, 9,542 BPS-FH and 13,470 Isophonics major/minor chord events remain; sequences shorter than three tokens are removed and consecutive duplicates compressed.Each corpus is augmented through all 12 pitch-class transpositions, balancing root distributions within each quality; Table 1 summarizes original-corpus statistics.Source: §3.1 · p. 2, Table 1 · p. 2
3.2 Embedding Model and Training Setup
- The skip-gram model represents each chord as a discrete root_quality token and predicts surrounding context tokens from a center token.Training uses a context window of 5, five negative samples, 64-dimensional embeddings, fixed random seeds, and single-threading.Source: §3.2 · p. 2
- The augmented setting trains for 50 epochs on the 12-fold transposed corpus, while the original setting trains for 600 epochs to approximately match total training-pair exposure.Two random initializations are trained per setting, and metrics are generally averaged across seeds.Source: §3.2 · p. 2
4. TONAL STRUCTURE IN LEARNED CHORD EMBEDDINGS
Across four analyses, augmented embeddings show more regular circle-of-fifths structure, stronger transposition consistency and key separation, and more interpretable key-directed shifts than non-augmented models.
4.1 Circle-of-Fifths Structure
- Augmented models show stronger circle-of-fifths structure on both datasets; BPS-FH has perfect cyclic ordering for major and minor, with Isophonics showing the same pattern particularly in minor.Source: Table 2 · p. 3, §4.1 · p. 3
4.2 Transposition Equivariance
- Augmented models achieve lower Procrustes error and higher neighbor overlap across datasets, while original models retain partial local consistency despite weak global alignment.Augmented training statistics are transposition-symmetric by construction, so some consistency is built into the data.Source: §4.2 · p. 3, Table 3 · p. 3
4.3 Emergent Key Structure
- Augmentation strengthens key separation across datasets and modes, especially minor; shuffling controls suggest broad tonal structure depends more on sequence-level co-occurrence than exact chord order.Source: Table 4 · p. 3, §4.3 · pp. 3–4
4.4 Tonal Region Steering
- At α = 2.0, augmented models on both datasets produce larger target-directed shifts and higher target-hit and source-escape rates, indicating more coherent key-related geometry.Source: §4.4 · p. 4
5. RELATION TO TONAL TENSION
The section asks whether learned embedding distances can express tonal tension, using key-relative and local chord-context measures. It tests their transposition stability, correspondence to matched TIV quantities, and alignment with human ratings.
5.1 Tension Measures and Transposition Stability
- Tkey weights chord-to-key-centroid distances by a prefix-based soft key distribution, while Tctx measures distance from the prior three-chord centroid; Tcomb combines both.The key distribution uses the chord-prefix mean, capped at the first eight chords.Source: §5.1 · pp. 4–5
- Augmentation substantially improves Tkey transposition stability on both datasets; Tctx retains higher curve-shape correlation, especially on BPS-FH, with Tcomb between them.The evaluation compares curves after non-identity transpositions using relative ℓ2 error and correlation.Source: §5.1 · p. 5, Table 6 · p. 5
5.2 Relation to TIV-Based Tonal Tension Components
- Pearson correlations around 0.71–0.73 and Spearman correlations around 0.69–0.70 link BPS-FH contextual distances to matched TIV quantities in both model families.Across both datasets, augmented models show a clearer advantage for key-relative correspondence, while contextual correspondence remains relatively similar across model families.Source: Table 7 · p. 5, §5.2 · p. 5
5.3 Relation to Human Tension Ratings
- r = 0.473, ρ = 0.531, RMSE = 0.189 on all 12 progressions; leave-one-progression-out scores were r = 0.406, ρ = 0.456, RMSE = 0.199.The measure reached r ≥ 0.6 on 7 of 12 progressions in both evaluations. Progression A showed broad model agreement, B closely followed participant ratings, and C exposed a mismatch likely tied to factors beyond chord labels.Source: Table 8 · p. 6, §5.3 · p. 6
- The human-rating analysis is diagnostic rather than a full predictor because chord labels omit inversion, realized bass motion, voicing, rhythmic salience, and non-chord tones.The dynamic measure combines embedding-derived key-relative and contextual distances with repetition and local-resolution terms.Source: §5.3 · pp. 5–6
6. CONCLUSION AND FUTURE WORK
Skip-gram embeddings recover interpretable tonal organization and tension-related structure, with generally stronger and clearer organization under transposition augmentation. The tension evidence is diagnostic rather than a complete perceptual model.
6. CONCLUSION AND FUTURE WORK
- Embeddings recover circle-of-fifths organization, transposition equivariance, key-centered separation, tonal-region steering, and tension-related structure.Source: §6 · p. 6
- Transposition augmentation generally strengthens these properties, yielding clearer and more stable harmonic organization.Source: §6 · p. 6
- Embedding distances correspond to matched TIV quantities, while a lightweight dynamic measure moderately agrees with human ratings; tension analysis remains diagnostic, not a complete perceptual model.Source: §6 · p. 6
- Future work should test larger heterogeneous corpora such as ChoCo, expand beyond triads, and add contextual or note-level information.Source: §6 · p. 6
7. AI USAGE STATEMENT
An AI-based language assistant supported manuscript text revision and wording refinement.
7. AI USAGE STATEMENT
- An AI-based language assistant supported text revision and wording refinement during manuscript preparation.Source: §7 · p. 7