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The remainder of this text will always use the term variable in this latter meaning (as long as not explicitly noted otherwise). Sometimes, the term constrained variable will be used to explicitly denote a variable in the context of constraint programming. A constrained variable is a variable which has a domain, that is a set of values it may take in a solution. g. the domain of all real numbers in some interval). However, in the remainder of this text, the domain of a variable is usually a finite set of possible variable values.

In the example in Fig. 8, one ‘dimension’ expresses the sequence of objects in a part, the other the set of objects in a measure. Other examples include grouping, harmonic information, or motif boundaries. Each of these ‘dimensions’ could even be hierarchically nested. For example, the metric ‘dimension’ could nest notes recursively in beats, measures, and higher-level accent groups. The harmonic ‘dimension’, on the other hand, could represent prolongational trees created by Schenkerian analysis.

When listening to music, we perceive various aspects such as rhythm, harmony, voice leading or instrumentation simultaneously. During the compositional process as well, the composer builds up a complex network of relationships between all the musical elements, and observes these elements from various viewpoints more or less simultaneously. Similarly, the formalisation of a task as complex as composition is greatly simplified when the task is stated in a modular way. When the task description is broken down into rhythmic rules, melodic rules, rules on the harmony etc.

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