State-space model solves figured-bass music constraints efficiently
A State-Space Model of Figured-Bass Realization: Local Constraints, Coupled Voices, and Polynomial-Time Solvability
SoundData Structures and Algorithms
Summary
Figured-bass realization is about choosing notes that fit musical rules both within each chord and between chords. The authors created a clear mathematical way to represent these choices and rules for a four-part music problem. They turned the problem into a path through a graph and showed it can be solved quickly with computers for fixed voices and rules. Their work helps explain when and how certain music combinations can be legally and optimally selected, though it does not capture every musical judgment.
What this means in practice
- •For music software developers: Design automatic music arrangement tools that efficiently realize musical constraints for four-part harmony tasks.
- •For digital audio workstation engineers: Integrate constraint-based harmony checking and suggestion features in DAWs using polynomial-time solvable models.
A theory result. No direct application yet.
Authors
Evan Unit Lim
Abstract
Figured-bass realization can be described as a sequence of choices constrained both within each sonority and between successive sonorities. This paper gives an explicit mathematical model of a restricted, examination-style four-part realization problem. Pitch spelling, range, chord membership, doubling, omission, spacing, crossing, overlap, melodic motion, consecutive perfect intervals, and selected resolution requirements are expressed as predicates. We distinguish hard constraints from optional preference costs. Four labeled notes are represented visually as the vertices of a quadrilateral and computationally as one ordered voicing state. Legal progressions become paths through a layered graph. We prove that feasibility and minimum-cost realization are polynomial-time problems for a fixed number of voices with explicit finite note domains and fixed local rules. For fixed ranges, a fixed note alphabet, and adjacent-event rules, the number of graph operations is linear in the number of events. Worked two-, four-, and eight-beat examples illustrate legality, optimization, and the failure of a greedy choice. The result concerns the stated formal model; it is not a claim that every musical judgment is captured by local predicates.