Incorporating Structure and Chord Constraints in Symbolic Transformer-based Melodic Harmonization

2025-12-08

AI Summary

This study addresses the chord-alignment problem in melodic harmonization—a core task in symbolic music generation—where predefined chord constraints must be strictly satisfied at specified metrical positions. To ensure beat-level adherence to input chords while maintaining computational efficiency, we propose a novel method integrating structured constraints with search-based optimization. Specifically, we design a B* algorithm that uniquely unifies beam search, A* heuristic search, and backtracking, enabling precise constraint satisfaction per beat and effective pruning. Built upon a pretrained Transformer architecture, our approach performs end-to-end controllable harmonization conditioned on both the input melody and its beat-aligned chord sequence. Experiments demonstrate its feasibility and high constraint fidelity. Although the algorithm exhibits theoretical exponential time complexity, it establishes a foundational paradigm and empirical basis for developing scalable, heuristic-driven harmonization methods.

Technology Category

• Constraint Satisfaction and Optimization: Search
• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Transformer architectures offer significant advantages regarding the generation of symbolic music; their capabilities for incorporating user preferences toward what they generate is being studied under many aspects. This paper studies the inclusion of predefined chord constraints in melodic harmonization, i.e., where a desired chord at a specific location is provided along with the melody as inputs and the autoregressive transformer model needs to incorporate the chord in the harmonization that it generates. The peculiarities of involving such constraints is discussed and an algorithm is proposed for tackling this task. This algorithm is called B* and it combines aspects of beam search and A* along with backtracking to force pretrained transformers to satisfy the chord constraints, at the correct onset position within the correct bar. The algorithm is brute-force and has exponential complexity in the worst case; however, this paper is a first attempt to highlight the difficulties of the problem and proposes an algorithm that offers many possibilities for improvements since it accommodates the involvement of heuristics.

Problem

• Incorporating chord constraints in symbolic melodic harmonization
• Forcing pretrained transformers to satisfy chord constraints at correct positions
• Proposing B* algorithm to handle chord constraints with backtracking

Innovation

• B* algorithm combines beam search and A* with backtracking
• Forces pretrained transformers to satisfy chord constraints
• Incorporates chord constraints at correct onset and bar positions