Introduction

Harmonic tension is the perceptual force that drives expectation and resolution in tonal music. While computational systems have modeled tension [1, 2], applying these models across both polyphonic and monophonic corpora remains challenging.

We contribute: (1) a multi-signal tension combination formula incorporating dynamic melodic leaps; (2) the Structural Tension Index (STI) to summarize peak tension position; and (3) a cross-corpus analysis, released as an open-source reproducible analytical pipeline.

Methods

Corpora

We analyze three bundled music21 corpora: 100 Bach chorales, a pilot sample of 22 Beethoven pieces (parsed as .mxl), and 31 essenFolksong melodies. We acknowledge the Beethoven sample size ($N=22$) is too small for definitive corpus-level claims; the framework must be run on larger external datasets in future work.

Tension Signals

Chord dissonance $D_b$: We assign roughness weights to interval classes using a discretized version of Huron's (1994) roughness model [3].

Vertical Density Change $\mathit{HR}_b$: We operationalize "harmonic rhythm" using music21's chordify() function. We acknowledge a significant musicological limitation: chordify() captures every note onset as a chord change, effectively measuring rhythmic density rather than functional harmonic progression. We therefore strictly define this signal as vertical density change.

Melodic Leap Tension $L_b$: Computes a dynamic $L_b$ tracking the normalized interval jump size between beats, preventing monophonic tension from collapsing to zero.

Combined Tension and STI

Per-beat tension is:

$$T_b = w_1 \cdot \hat{D}_b + w_2 \cdot \mathit{HR}_b + w_3 \cdot L_b$$

Uncalibrated Priors: The weights ($w_1=0.5, w_2=0.3, w_3=0.2$) are currently uncalibrated heuristic priors. Future work must fit these parameters empirically against ground-truth human perceptual data (e.g., Krumhansl's tension annotations) to validate the metric scientifically.

Recognizing that a single argmax is statistically fragile, we apply a 4-beat rolling average to smooth the tension curve and contextualize the STI using $k$-means archetype clustering to capture the overall structural "arc."

Results

Cross-Genre Normalization: There is a significant magnitude mismatch in raw tension values (Folk = 0.038 vs. Bach = 0.469). Future iterations must apply corpus-specific z-score normalization to the tension signals before combining them to ensure valid cross-genre comparability.

Statistical Inference: The $k=4$ KMeans clustering reveals that polyphonic works reflect declining and arch profiles, while monophonic folk songs cluster into a flat plateau. Because the STI (peak position) is topologically arbitrary for a flat plateau, the metric is invalid for the folk corpus. Consequently, we exclude the folk corpus from statistical testing. An independent samples t-test between the valid polyphonic corpora (Bach vs. Beethoven) confirms a significant difference in peak position ($t(120) = 3.42, p < 0.001$), though this requires validation on a larger Beethoven sample.

Conclusion

We present a computational framework for mapping musical tension arcs. By identifying limitations in heuristic weighting, normalization, and rhythmic density capture, this study defines the necessary constraints and calibration steps required before deploying structural tension metrics across heterogeneous musical genres.

References

[1] Lerdahl, F., & Jackendoff, R. (1983). A Generative Theory of Tonal Music. MIT Press.
[2] Herremans, D., & Chew, E. (2017). MorpheuS: generating structured music with constrained patterns and tension. IEEE Transactions on Affective Computing, 9(4), 510-523.
[3] Huron, D. (1994). Interval-class content in equally tempered pitch-class sets: Common scales exhibit optimum tonal consonance. Music Perception, 11(3), 289-305.