💻 computer science

Tonnetz-Driven Graph Wedgelet for Harmonic Complexity Reduction in Music Scores

This paper proposes a method for reducing the harmonic complexity of symbolic music scores by constructing a heterogeneous graph of musical elements and compressing the piano subgraph using an adaptive greedy algorithm that partitions notes into wedges based on a six-dimensional Tonnetz embedding to generate simplified, playable scores while preserving harmonic relationships.

Original authors: Emmanuel Caronna, Elisa Francomano, Silvia Licciardi

Published 2026-07-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Emmanuel Caronna, Elisa Francomano, Silvia Licciardi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a massive, intricate LEGO castle representing a piece of music. Every single brick is a note, and they are all stacked in a specific, complex pattern. Now, imagine you want to shrink this castle down to fit in your pocket without losing the "soul" of the building. You can't just smash it randomly, or it won't look like a castle anymore. You need a smart way to group the bricks so that when you rebuild it, it still looks and sounds like the original, just simpler.

That is exactly what this paper does, but instead of LEGOs, it uses music scores, and instead of hands, it uses a clever mathematical tool called a graph.

The Problem: Too Many Bricks, Not Enough Space

Music scores, especially the piano parts that accompany a singer, are often full of repetition. Think of a song where the piano plays a fancy, rolling pattern over and over again. To a computer, every single note in that rolling pattern is a separate, unique brick. This creates a "graph" (a map of connections) that is huge and messy.

The authors argue that previous ways of simplifying music were like trying to flatten a 3D sculpture into a flat sheet of paper. They either ignored the musical relationships between notes or treated the music like a simple list of sounds, missing the deep, harmonic connections that make music sound "right."

The Solution: The "Harmony Map" (Tonnetz)

To fix this, the researchers built a special map called a Tonnetz. Imagine a standard clock face where the numbers 1 through 12 represent musical notes. On a normal clock, the distance between 1 and 2 is the same as 1 and 7. But in music, 1 and 7 (a perfect fifth) are best friends, while 1 and 2 are strangers.

The authors use a six-dimensional version of this map. Think of it as a magical, multi-layered playground where notes that sound good together are physically close to each other, and notes that clash are far apart. This is crucial because it ensures the computer understands harmony, not just the order of notes.

The Magic Trick: The "Wedge" Slicer

Here is the core of their invention: the Graph Wedgelet.

Imagine you have a giant, messy pile of clay (the music graph). You want to carve it into big, smooth chunks called wedges. The goal is to make each chunk so uniform that you can describe the whole chunk with just one single number (the average of the notes inside it).

The paper introduces a "Fully Adaptive Greedy" algorithm. This is a fancy way of saying the computer plays a game of "hot and cold":

It looks at the whole pile of clay.
It asks, "If I cut this in half right now, which cut will make the two new piles the most similar to each other?"
It makes the cut.
It repeats this process, recursively splitting the piles until it has the number of chunks it needs.

The "cut" isn't random. It uses the Tonnetz map to ensure that notes grouped together are harmonically related. If a chunk contains a C-major chord, the computer keeps it together. It won't accidentally group a C with a note that sounds terrible next to it.

The Result: A Simpler, Playable Score

Once the music is chopped into these "wedges," the computer replaces every single note in a wedge with the average note of that group.

Think of it like this: If a wedge contains 50 notes that are all variations of a C-major chord, the computer says, "Okay, for this whole section, we'll just play the 'average' C."

The paper shows that this works beautifully. In their simulations, they took 70 real music scores from three different composers. They tested how much they could shrink the music (the compression ratio) before it sounded bad.

When they kept everything (100% of the notes), the result was perfect.
When they compressed it heavily (down to just a few wedges), the music became much simpler, but it still sounded like the original song.

The paper explicitly states that this method does not introduce new notes that weren't in the original song. It only picks from the notes the composer already wrote. This is a safety feature: the simplified score is always playable and readable by a human, and it never invents weird, alien notes that don't belong.

What This Isn't (and What It Doesn't Do)

It's important to know what this paper doesn't claim.

It doesn't claim to be a magic wand that can compress any file instantly. It specifically works on the piano accompaniment part of vocal-piano scores. It doesn't yet handle the melody or the lyrics in the same way.
It doesn't say this is the "perfect" compression method for all time. The authors admit that right now, they replace a whole wedge with a single, constant note. In the future, they hope to use more complex shapes to capture the movement inside a wedge, but for now, it's a flat, constant average.
The results are based on simulations and experiments on a specific set of 70 scores. They haven't tested this on every piece of music ever written, but the math checks out for the ones they tried.

The Bottom Line

The authors have built a tool that acts like a smart editor for music. It looks at a complex piano part, finds the hidden harmonic patterns, and groups similar notes together into "wedges." Then, it simplifies the score by replacing those groups with their average sound.

The result is a new, simpler version of the music that is easier to read and play, but still sounds like the original piece. It's like taking a high-resolution photo and turning it into a pixel-art version that still looks exactly like the subject, just with fewer pixels. The paper suggests this could be the first step toward software that automatically creates simplified arrangements for beginners, ensuring the music stays true to the composer's original vision.

Technical Summary: Tonnetz-Driven Graph Wedgelet for Harmonic Complexity Reduction in Music Scores

Problem Statement
Symbolic music scores, particularly those containing vocal and piano parts, exhibit significant structural redundancy in their accompaniment. Harmonic figurations, such as Alberti basses, arpeggiated chords, and ostinati, often repeat the same harmonic sequences over extended temporal spans. While existing compression methods for symbolic music exist—ranging from algorithmic information theory (e.g., Lempel-Ziv) to wavelet transforms on melodic signals and tokenization for neural processing—they suffer from specific limitations. Current approaches typically operate at the signal, sequential event, or symbolic token level. Crucially, they fail to simultaneously: (i) adapt to the geometry of the underlying graph representation of the score; (ii) account for harmonic distance rather than mere chromatic distance; and (iii) constrain the reconstruction to remain a valid musical score within the original scale and instrumentation. There is a need for a compression scheme that reduces harmonic complexity while preserving task-relevant information, note relations, and graph structure.

Methodology
The authors propose a structure-preserving compression scheme for the piano subgraph of vocal-pianistic scores, utilizing a Tonnetz-driven Graph Wedgelet framework. The methodology integrates graph signal processing with music theory as follows:

Graph Representation: The music score is modeled as a heterogeneous graph where nodes represent notes (melodic line, lyrics, accompaniment) and edges encode sequential, vertical, and cross-modal relationships. The compression focuses specifically on the piano subgraph ( $G_{piano}$ ).
Tonnetz Embedding: To ensure the compression respects harmonic relationships, musical notes are mapped to a six-dimensional vector space using a Tonnetz embedding. This embedding projects pitch classes onto a circle using three distinct integer factors ( $k_1, k_2, k_3$ ) to capture major thirds, minor thirds, and the circle of fifths. The Euclidean metric in this 6D space captures harmonic proximity rather than simple chromatic distance.
Binary Wedge Partitioning (BWP): The graph is partitioned into regions called "wedges" using a Fully Adaptive Greedy (FA-greedy) algorithm.
- The process constructs a binary tree by recursively splitting the graph.
- Splitting Criterion: At each step, the algorithm selects a wedge to split and a new center node ( $v_{m+1}$ ) to minimize the $L_2$ -error of the resulting partition within the 6D Tonnetz embedding.
- Connectivity: The splitting is based on shortest-path distance on the graph, ensuring that every resulting wedge remains a connected subgraph (musically contiguous).
Encoding and Decoding:
- Encoder: The FA-greedy algorithm generates a set of $M$ wedges based on a compression ratio $r$ . The signal within each wedge is approximated by the mean value of the notes in the Tonnetz embedding.
- Decoder: To reconstruct a playable score, the mean value of each wedge is quantized. The algorithm selects the nearest pitch class present in the original wedge from the set of pitch classes actually used in that region ( $PC(V_{vi})$ ). This ensures the reconstructed notes are drawn from the composer's original palette, preventing the introduction of foreign pitch classes.
Export: The reconstructed piecewise-constant MIDI signal is mapped back to the original MusicXML structure, preserving all non-pitch attributes (duration, dynamics, articulation, etc.) while updating the pitch values.

Key Contributions

Harmonic-Aware Compression: The paper introduces a compression method that explicitly minimizes error in a harmonic space (Tonnetz) rather than a chromatic or time-series space, ensuring that the reduced complexity reflects intrinsic harmonic relationships.
Graph-Adaptive Partitioning: By applying binary wedge partitioning trees to the score-graph, the method adapts to the specific topology of the musical accompaniment, maintaining the connectivity of musical fragments.
Valid Score Reconstruction: The proposed quantization strategy guarantees that the output is a valid, human-readable, and playable score that retains the original instrumentation and scale, avoiding the generation of "hallucinated" notes not present in the source material.
Integration of Frameworks: The work extends the graph wedgelet framework (previously applied to images and biomedical data) to the domain of symbolic musicology, combining it with multi-dimensional pitch embeddings.

Experimental Results
The authors evaluated the approach on a corpus of 70 symbolic music scores (MusicXML format) by three different composers.

Metric: The performance was measured using the Root-Mean-Square Error (RMSE) between the original Tonnetz-valued signal and the wedgelet approximation, computed vertex-wise.
Findings: The experiments demonstrated a trade-off between the compression ratio ( $r$ ) and reconstruction fidelity. As the ratio $r$ decreases (fewer wedges), the RMSE increases, indicating greater harmonic deviation. However, the visual analysis of reconstructed measures (Fig. 4) showed that even at lower ratios, the system preserves the original pitch for a significant portion of notes (green noteheads), with reassignments (yellow noteheads) occurring primarily where harmonic simplification is most effective. The method successfully generated simplified scores that remained recognizable as the original pieces.

Significance and Claims
The paper positions this work as a necessary step toward the automatic generation of simplified musical arrangements. The authors claim that their method establishes a "necessary condition" for this objective: ensuring that a compressed arrangement remains a valid musical score that is recognizable as the original piece. By preserving the harmonic structure and the graph topology, the method allows for the reduction of harmonic complexity without distorting the musical identity of the accompaniment. The authors modestly note that the current scheme operates on the homogeneous piano subgraph in isolation and that future work is required to extend the error metrics and splitting criteria to the full heterogeneous graph (including melody and lyrics) and to explore non-constant functions within wedges to capture intra-wedge harmonic motion.