Musical Harmonization with Constraints: A Survey

1998

Abstract I discuss three issues relevant to the use of constraints in musical AI applications: four-part harmony; serial composition; and music representation. I describe two small pieces of work in the former two, and suggest that the last constitutes an important challenge question for those interested in designing constraint systems for use in an AI/Music context.

Proceedings of the SMC Conferences, 2017

Music constraint systems provide a rule-based approach to composition. Existing systems allow users to constrain the harmony, but the constrainable harmonic information is restricted to pitches and intervals between pitches. More abstract analytical information such as chord or scale types, their root, scale degrees, enharmonic note representations, whether a note is the third or fifth of a chord and so forth are not supported. However, such information is important for modelling various music theories. This research proposes a framework for modelling harmony at a high level of abstraction. It explicitly represents various analytical information to allow for complex theories of harmony. It is designed for efficient propagationbased constraint solvers. The framework supports the common 12-tone equal temperament, and arbitrary other equal temperaments. Users develop harmony models by applying user-defined constraints to its music representation. Three examples demonstrate the expressive power of the framework: (1) an automatic melody harmonisation with a simple harmony model; (2) a more complex model implementing large parts of Schoenberg's tonal theory of harmony; and (3) a composition in extended tonality. Schoenberg's comprehensive theory of harmony has not been computationally modelled before, neither with constraints programming nor in any other way.

1998

Product configuration is when an artifact from a product family is assembled from a set of predefined components that can only be combined in certain ways. These ways are defined by configuration rules. The product developers inspect the configuration rules when they develop new configuration rules or modify the configuration rules set. The inspection of configuration rules is thereby an important activity to avoid errors in the configuration rules set. Several formulations of constraint satisfaction problems (CSPs) are proposed that facilitate the inspection of configuration rules in propositional logic (IF-THEN, AND, NOT, OR, etc.). Many of the configuration rules are so called production rules; that is, a configuration rule is an IF-THEN expression that fires when the IF condition is met. Several configuration rules build chains that fire during the product configuration. It is therefore important not only to inspect single configuration rules but also to analyze the effect of multiple configuration rules. Formulating the tasks as variations of the CSP can support the inspection activity. More specifically, we address the reformulation of configuration rules, testing of feature variant combinations, and counting of item quantities from an item set. The suggested CSPs are tested on industrial vehicle configuration rules for computational performance. The results show that the time for achieving results from the solving of the CSP is within seconds. Our future work will be to implement the various CSPs into a demonstrator that could be tested by product developers.

Computer Music Journal, 2010

2009

Harmonization with four voices is a musical problem which is subject to hard constraints, which absolutely need to be fulfilled, as well as to soft constraints, which preferably hold, but are not mandatory. In this paper, we model this problem as a valued constraint satisfaction problem (VCSP): costs are assigned to possible solutions based on the constraints they violate. We design an algorithm that finds a minimal-cost solution, thus solving the harmonization problem, and we present initial results obtained by this algorithm.

2000

Abstract Constraints programming allows the composer to synthesize a score by describing it. Arno is a program for computer assisted composition which extends Common Music (CM) by means of constraints programming using Screamer. In Arno parameters of CM elements in a CM container can be declared nondeterministically using finite domains—instead of single values. Constraints are expressed as predicates which test one CM object and restrict the actual values of its slots.

Proceedings of International Computer Music Conference (ICMC 2008), Belfast, UK, 2008

This paper studies how constraints are applied to the score in a musical constraint satisfaction problem (CSP). How can we control which variable sets in the score are affected by a given constraint? Our overall objective is to produce a highly generic music constraint system, where users can define a wide range of musical CSPs, including rhythmic, harmonic, melodic and contrapuntal problems. Existing systems provide constraint application mechanisms, which are convenient for specific cases, but lack generality and cannot be extended by users. As a result, complex sets of variables are hard to constrain in these systems. For example, constraining notes from different voices in a polyphonic setting (e.g., with harmonic constraints) is impossible in most systems. We propose an approach which combines convenience with full user control: higher-order constraint applicators. A constraint is a first-class function, while a constraint applicator is a higher-order function which traverses the score in order to apply a given constraint to variable sets. This text presents constraint applicators suitable for a many musical CSPs, and reproduces important mechanisms of existing systems. Most importantly, users can define their own constraint applicators with this approach.

During the last decades, several methodologies have been proposed for the harmonization of a given melody with al-gorithmic means. Among the most successful are method-ologies that incorporate probabilistic mechanisms and sta-tistical learning, since they have the ability to generate har-monies that statistically adhere to the harmonic character-istics of the idiom that the training pieces belong to. The current paper discusses the utilization of a well–studied probabilistic methodology, the hidden Markov model (HMM), in combination with additional constraints that incorporate intermediate fixed–chord constraints. This work is moti-vated by the fact that some parts of a phrase (like the ca-dence) or a piece (e.g. points of modulation, peaks of ten-sion, intermediate cadences etc.) are characteristic about the phrase's or piece's idiomatic identity. The presented methodology allows to define and isolate such important parts/functions and include them as constraints in a proba...

2007