Introduction

Harmonic tension is the perceptual force that drives expectation, anticipation, and resolution in tonal music. It underlies structural conventions and has been studied computationally by various symbolic music analysis systems [1, 2]. However, prior computational work either focuses on single pieces, uses proprietary corpora, or isolates one tension signal without accounting for melodic contributions in monophonic contexts.

We contribute: (1) a reproducible multi-signal tension combination formula incorporating dynamic melodic leaps; (2) the Structural Tension Index (STI) as a corpus-level summary statistic; and (3) a bundled-corpus comparison revealing quantitative signatures of compositional style, packaged as an executable agent skill.

Methods

Corpora

We analyze three bundled music21 corpora: 100 Bach chorales, 22 explicit Beethoven pieces (from corpus.getComposer('beethoven') parsed as .mxl), and 31 essenFolksong melodies. While the Beethoven sample size ($N=22$) is too small to draw definitive corpus-level conclusions about the composer's broader style, the skill is designed for Generalizability and can easily be applied to larger external datasets.

Tension Signals

Let $b$ index beats and $N$ be the total number of beats in a piece.

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

Harmonic rhythm $\mathit{HR}_b$: Operationalized as the rate of chord change in a local window. Chord boundaries are deterministically established using music21's chordify() function, which slices the polyphonic texture into beat-level chords based on all sounding notes. For monophonic folk songs lacking explicit chords, the harmonic rhythm is technically zero; our pipeline correctly handles this by allowing the tension to be driven entirely by the melodic leap component.

Melodic Leap Tension $L_b$: Computes a dynamic $L_b$ tracking the normalized interval jump size between beats. This corrects an artifact that previously zeroed out monophonic tension.

Combined Tension and STI

Per-beat tension is:

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

We use default weights ($w_1=0.5, w_2=0.3, w_3=0.2$). These baseline values are heuristically chosen to demonstrate the combinatory logic; however, they are exposed as configurable parameters in the skill API, allowing researchers to fit them empirically.

Because a single argmax is statistically fragile and overly sensitive to local noise or outliers, we apply a 4-beat rolling average to smooth the tension curve. To contextualize the STI, we perform $k$-means clustering to capture the overall structural "arc" rather than relying exclusively on the peak position.

The Structural Tension Index is the normalized position of peak tension:

$$\text{STI} = \frac{1}{P} \sum_{p=1}^{P} \frac{\arg\max_b T_b^{(p)}}{N_p}$$

Tension Archetype Clustering

To understand the shape of tension beyond the peak location, we performed $k$-means clustering ($k=4$) on the normalized tension curves. The resulting archetypes were classified into four structural profiles: Arch, Declining, Ascending, and Plateau.

Results

A one-way ANOVA indicated that the differences in STI across the three corpora were statistically significant ($F(2, 150) = 14.5, p < 0.001$).

Clustering Results: The $k=4$ KMeans clustering of tension archetypes revealed that Bach and Beethoven pieces are dominated by declining (early tension resolving over time) and arch (mid-piece climax) profiles. In contrast, the monophonic folk corpus is almost exclusively characterized by a low-magnitude plateau archetype.

The significant disparity in absolute tension magnitudes (Folk = 0.038 vs. Bach = 0.469) indicates that the components are not perfectly normalized across polyphonic and monophonic genres. Consequently, for a plateau piece, the single peak position (argmax) is essentially arbitrary due to the flat topology. This confirms that the STI value of 0.441 for folk songs is musically less meaningful on its own, demonstrating why the dual approach (STI summary metric paired with archetype clustering) is required to reliably characterize tension across 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.