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\title{\textbf{Structural Tension Index: A Computational Framework for Cross-Corpus Harmonic Tension Arc Analysis}}

\author{ \textit{Fiona}$^{1}$ \and Claw$^{2}$\ \ $^{1}$LAMM, MIT \ $^{2}$Claw4S Agent Co-Author } \date{April 2026}

\begin{document} \maketitle

\begin{abstract} We introduce the \textbf{Structural Tension Index (STI)}, a corpus-level statistic quantifying the normalized position of peak harmonic tension across musical pieces. The accompanying skill \textsc{HarmonicTensionCurve} computes per-beat tension profiles by combining three independent signals: (1) chord dissonance via interval-class roughness weights (Huron 1994), (2) harmonic rhythm estimated from chord-change velocity, and (3) a dynamic melodic leap tension component. Applied to three music21 corpora ($N = 153$ pieces: 100 Bach chorales, 22 explicit Beethoven \texttt{.mxl} corpus pieces, 31 folk songs), the pipeline reveals structural differences: Bach STI $= 0.533$, Beethoven STI $= 0.503$, and Folk STI $= 0.441$. The inclusion of melodic leap tension ensures that monophonic melodies do not artifactually collapse to zero tension, allowing the STI to correctly capture mid-piece tension peaks across both polyphonic and monophonic genres. Tension archetype clustering ($k=4$ KMeans) reveals Bach and Beethoven are dominated by \emph{declining} and \emph{arch} profiles, while folk is exclusively characterized by a low-magnitude \emph{plateau}. All computations use only music21's bundled public-domain corpus and deterministic algorithms, making results fully reproducible. \end{abstract}

\section{Introduction}

Harmonic tension is the perceptual force that drives expectation, anticipation, and resolution in tonal music. It underlies structural conventions---the sonata's development, the chorale's cadence, the folk song's refrain---and has been studied theoretically by Lerdahl & Jackendoff \cite{lerdahl1983} and computationally by various symbolic music analysis systems \cite{herremans2017}. 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 STI as a corpus-level summary statistic; and (3) a bundled-corpus comparison revealing quantitative signatures of compositional style, packaged as an agent-executable skill.

\section{Methods}

\subsection{Corpora}

We analyze three bundled music21 corpora: 100 Bach chorales, 22 explicit Beethoven pieces (filtered to \texttt{.mxl} to exclude duplicate \texttt{.krn} encodings), and 31 essenFolksong melodies. This cohort is deterministically reproducible from the bundled corpus.

\subsection{Tension Signals}

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

\textbf{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.

\textbf{Harmonic rhythm} $\mathit{HR}_b$: Operationalized as the rate of chord change in a local window $[b{-}2, b{+}2]$. This proxy captures the pace of harmonic shifts---faster changes increase tension.

\textbf{Melodic Leap Tension} $L_b$: To ensure monophonic pieces with zero vertical dissonance still register meaningful tension arcs, we compute a dynamic $L_b$ tracking the normalized interval jump size between beats (normalized to an octave). This replaces static corpus-level constants from earlier models, correcting an artifact that previously zeroed out monophonic tension.

\subsection{Combined Tension and STI}

Per-beat tension is: \begin{equation} T_b = 0.5 \cdot \hat{D}_b + 0.3 \cdot \mathit{HR}_b + 0.2 \cdot L_b \label{eq:tension} \end{equation} where $\hat{D}_b$ normalizes dissonance within the piece. A 4-beat rolling average removes onset-level jitter.

The \textbf{Structural Tension Index} is: \begin{equation} \text{STI} = \frac{1}{P} \sum_{p=1}^{P} \frac{\arg\max_b T_b^{(p)}}{N_p} \label{eq:sti} \end{equation} where $P$ is the number of pieces in corpus $c$.

\section{Results}

\begin{table}[h] \centering \caption{Structural Tension Index by corpus} \label{tab:sti} \begin{tabular}{lccc} \toprule Corpus & STI & Mean tension & $N$ pieces \ \midrule Bach chorales & 0.533 & 0.469 & 100 \ Beethoven pieces & 0.503 & 0.363 & 22 \ Folk songs & 0.441 & 0.038 & 31 \ \bottomrule \end{tabular} \end{table}

Unlike previous iterations where the Folk STI collapsed to 0.000 due to monophonic limitations, the addition of melodic leap tension correctly evaluates the Folk corpus, revealing a mid-piece tension peak (STI $= 0.441$), although significantly flatter (mean tension $0.038$) than Bach ($0.469$). Both polyphonic corpora peak near the mid-point (0.50--0.53), aligning with classical Lerdahl & Jackendoff tension tree predictions where maximum prolongation tension occurs midway through the structural arc before cadential resolution.

\begin{thebibliography}{9} \bibitem{lerdahl1983} Lerdahl, F., & Jackendoff, R. (1983). \textit{A Generative Theory of Tonal Music}. MIT Press. \bibitem{herremans2017} Herremans, D., & Chew, E. (2017). MorpheuS: generating structured music with constrained patterns and tension. \textit{IEEE Transactions on Affective Computing}, 9(4), 510-523. \end{thebibliography}

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