vega-mir: An information-theoretic Python toolkit for symbolic music, with applications to harmonic graphs and rubato spectra

This paper introduces *vega-mir*, an open-source Python toolkit for symbolic music analysis featuring nine information-theoretic metrics, and demonstrates its utility through case studies revealing a correlation between harmonic graph centrality and harmonic distance across composers, as well as evidence that Glenn Gould's rubato is characterized by structured periodicity rather than metronomic rigidity.

The Gist

Imagine you have a massive library of sheet music from different composers and performers. For a long time, music researchers have tried to understand these libraries by taking simple "snapshots"—like counting how often a composer uses a specific note or measuring the average speed of a performance. But these snapshots often miss the bigger picture, like the flow of a conversation or the rhythm of a heartbeat.

This paper introduces vega-mir, a new, open-source "toolbox" for computer scientists and musicologists. Think of it as a Swiss Army knife that comes pre-loaded with nine specific mathematical tools designed to analyze music written as symbols (like sheet music or digital codes) rather than sound waves.

Here is a breakdown of what the paper actually does, using simple analogies:

1. The Toolbox (The Library)

Before this tool, if a researcher wanted to analyze music, they had to build their own measuring tape, their own scale, and their own calculator for every single project. It was messy and hard to compare results.

vega-mir is like a standardized, pre-calibrated kit. It bundles nine different "metrics" (ways of measuring) into one clean package.

• Three of these tools were already used in a previous study (called "Cygnus") to analyze thousands of piano recordings.
• Four are new "sanity checks" that the authors tested on a small group of composers to make sure they work correctly.
• Two are brand new tools that the authors use in this paper to dig deeper than ever before.

2. Case Study A: The "Harmonic Map" (Chord Transitions)

The first new tool looks at how chords move from one to another. Imagine a city map where every intersection is a musical chord.

• The Old Way: Researchers used to just count how many cars (chords) passed through each intersection. They knew which intersections were busy, but not how the traffic flowed between them.
• The New Way (vega-mir): This tool builds a full traffic map. It calculates a "gravity center"—a specific chord that acts like the main hub of the city, pulling the most traffic.
• The Discovery: The authors analyzed 14 famous composers (like Bach, Mozart, and Beethoven). They found that for most composers, the "gravity center" wasn't the home chord (the tonic), but a neighboring chord (the supertonic).
  • Analogy: It's like realizing that in a city, the most important hub isn't City Hall (the home), but the main train station (the neighbor) because that's where all the connections happen.
  • They also found that this "hub" location correlates with how different a composer's music sounds from others, but the type of hub (major vs. minor) doesn't tell the whole story.

3. Case Study B: The "Rubato Radar" (Tempo Changes)

"Rubato" is when a musician speeds up or slows down slightly for emotional effect. The old way of measuring this was to take the average speed of the whole performance and say, "This person is fast," or "This person is slow."

• The Problem: This is like judging a runner only by their average speed. It misses whether they are sprinting in bursts, jogging steadily, or drifting slowly.
• The New Way (vega-mir): This tool acts like a weather radar. Instead of just measuring wind speed, it looks at the pattern of the wind. Is it a steady breeze? A sudden gust? A rhythmic wave?
• The Discovery: The authors studied three famous pianists playing Bach: Glenn Gould, András Schiff, and Sviatoslav Richter.
  • The Cliché: People often say Glenn Gould plays like a "metronome" (perfectly robotic) because his average speed changes very little.
  • The Reality: The radar showed that Gould isn't robotic; he is just structured. While Schiff and Richter let their tempo drift freely (like a loose cloud), Gould's tempo changes in a very specific, rhythmic pattern (like a heartbeat).
  • The Twist: Gould actually had the most rhythmic structure (highest "periodicity") of the three. His "rubato" was small in size but very organized in time. The old "average speed" measurement hid this fact completely.

4. Why This Matters

The paper isn't claiming to discover new laws of physics or music theory. Instead, it's about consolidation.

• It takes complex math that usually requires a PhD to implement and turns it into simple, one-line commands anyone can use.
• It proves that looking at the structure of music (how chords connect, how tempo patterns repeat) reveals hidden details that simple averages miss.
• It provides a shared language so that different researchers can compare their results without arguing about who used the right calculator.

In short: The authors built a better microscope for music. They used it to show that a famous pianist isn't a robot, but a rhythmic architect, and that the "hubs" of musical harmony are often different than we thought. All of this is now available for anyone to use in their own research.

Technical Summary

Technical Summary of vega-mir: An information-theoretic Python toolkit for symbolic music

Problem Statement
While information-theoretic descriptors (e.g., Shannon entropy, Kullback-Leibler divergence, Zipfian fits) have become standard for the quantitative analysis of symbolic music, their implementation remains fragmented. Current research typically requires "stitching together" disparate calls to scipy.stats, hand-rolled smoothing algorithms, and ad-hoc network analysis code. This fragmentation results in a lack of shared APIs, test suites, or reference values, hindering reproducibility and cross-study validation. There is a need for a consolidated, tested library that bundles these metrics with consistent defaults and reference values for the music information retrieval (MIR) and computational musicology communities.

Methodology
The paper introduces vega-mir, an open-source Python library (v0.0.1) designed to operate on symbolic music inputs (scale degree sequences, chord-transition graphs, tempo curves). The library is built upon numpy, scipy, networkx, mido, and pretty_midi, and follows a rigorous validation strategy:

1. Analytic Anchors: Metrics are unit-tested against theoretical ground truths (e.g., uniform distributions yielding specific entropy values, white noise yielding specific fractal dimensions).
2. Parity Checks: Implementations are verified for exact parity against the upstream Cygnus pipeline (Jalbert-Desforges 2026).
3. API Design: The library exposes a three-layer API for each metric: a primitive function for normalized vectors, a counts-based convenience function with Laplace smoothing, and a sequence-based convenience function for chord/scale-degree inputs.

The paper catalogs nine metrics divided into four domains:

• Information Theory: Shannon entropy and Kullback-Leibler (KL) divergence (previously deployed in Cygnus).
• Rank-Frequency: Zipfian fits for marginal and transition distributions (previously deployed in Cygnus).
• Inequality: Multi-dimensional Gini coefficient.
• Networks: Network analysis on chord-transition graphs (PageRank, clustering, modularity).
• Time-Series/Signal Processing: Chi-squared stationarity tests, Higuchi fractal dimension, spectral rubato analysis, and interval distribution fitting.

Key Contributions

1. Library Consolidation: The primary contribution is the consolidation of nine metrics behind a single, tested, citable API with a shared set of reference values, lowering the activation cost for distributional analyses.
2. Case Study 1: Harmonic Network Signatures: A network analysis of chord-transition graphs for 14 MAESTRO composers ($N \ge 10$ pieces). The study constructs directed weighted graphs of scale-degree transitions and analyzes node-level (PageRank) and graph-level descriptors.
3. Case Study 2: Rubato Spectral Signatures: A spectral analysis of tempo curves (rubato) across a 247-piece Bach corpus performed by three pianists (Schiff, Gould, Richter). The study utilizes a spectral classifier to categorize rubato into four types (metronomic, free, quasi-periodic, periodic) based on power spectral density and periodicity ratios.

Results

• Harmonic Networks:

  • Graph Saturation: At the 1% transition-probability threshold, the harmonic transition graphs for all 14 composers are saturated (retaining ~120 of 210 possible edges) and exhibit small-world properties.
  • Gravity-Centre Identity: The PageRank "gravity centre" (the scale degree with the highest PageRank) varies across composers, taking five distinct values (e.g., Major Tonic I for Mozart; Supertonic II for Bach, Haydn, etc.). However, this categorical identity does not correlate with marginal KL distance ($\rho = 0.13, p = 0.21$).
  • Continuous Correlation: The continuous PageRank value of the gravity-centre node correlates with the marginal KL distance ($\rho = 0.61$, Spearman, $N=14$). This correlation is driven almost entirely by the single PageRank feature; other network descriptors (density, clustering) are quasi-uniform across the corpus.
  • Leverage: The correlation is sensitive to individual composers; removing Mendelssohn reduces the correlation significantly, indicating the result is a qualitative observation of ordering rather than a statistically stable estimate.

• Rubato Spectra:

  • Inversion of Cliché: Contrary to the scalar interpretation that Glenn Gould's low standard deviation ($\sigma$) implies "metronomic" playing, spectral analysis reveals Gould has the highest periodicity ratio (0.293) among the three pianists, significantly higher than Schiff (0.204) and Richter (0.257).
  • Temporal Colour: Gould's rubato is characterized as "small in amplitude but structured in time," with a median dominant period of 66 beats, compared to 102 (Schiff) and 104 (Richter).
  • Diachronic Analysis: In a comparison of Gould's 1955 vs. 1981 Goldberg Variations, the 1981 recording shows a significant reduction in rubato amplitude ($\Delta\sigma = -2.39$ BPM) but no significant shift in spectral colour (periodicity ratio change is non-significant). The 1981 performance is a "quieter" version of the same structural rubato, not a restructured one.
  • Robustness: The finding that Gould holds the majority of "periodic" classifications survives threshold sensitivity analysis (17/17 perturbation runs).

Significance and Claims
The paper explicitly frames vega-mir as a consolidation, not a discovery. The mathematical foundations of the nine metrics are decades old; the contribution lies in providing a "single tested API with consistent defaults and a shared set of reference values."

The authors claim that the library enables:

1. Reproducibility: By offering a shared test suite and reference values, results can be cross-validated across studies.
2. Signal Recovery: The case studies demonstrate that information-theoretic descriptors beyond marginal statistics (specifically network topology and spectral colour) recover signals that scalar summaries discard. For instance, the scalar mean of rubato obscures the "temporal colour" of performance, while network analysis reveals that the "gravity centre" of harmonic flow is a continuous descriptor correlated with marginal divergence, distinct from its categorical identity.
3. Accessibility: The library lowers the barrier to entry for researchers needing distributional analyses without reimplementing underlying mathematics.

The paper maintains modesty regarding statistical claims, noting that the network correlations are qualitative observations subject to high leverage from individual composers, and that four of the nine metrics (Gini, stationarity, Higuchi, intervals) are currently validated only on small datasets and await full corpus-scale empirical deployment in future work.