Divergent Production in Music Composition: A Computational Model

Solving a problem creatively means finding a good solution that also stands apart from previous ones. Creative solutions are defined mainly by two characteristics: appropriateness and originality. An appropriate solution satisfies the goal of the problem and is useful; it must also be coherent, with no incompatibilities between parts, and capable of being carried out. An original solution is different from what has come before — unique, novel, and somewhat unpredictable.

J.P. Guilford proposed that the mind's ability to generate creative solutions arises from what he called divergent production, a concept related to Edward De Bono's lateral thinking. This ability allows the generation of many different solutions to the same problem. It applies in domains where creative solutions cannot be judged simply as right or wrong, and instead are evaluated on originality and appropriateness. In contrast, convergent production produces the single correct answer for problems that have defined right and wrong solutions.

Although four main perspectives on creativity exist — the creative person, the creative environment, the creative product, and the creative process — the latter two receive the most attention in the literature.

Among models of the creative process, those related to problem solving are of particular interest. Graham Wallas proposed that creativity involves four sequential stages: preparation, incubation, illumination (or insight), and verification. Preparation involves identifying a problem and acquiring the knowledge needed to solve it. Incubation is the process of reassembling past knowledge into new structures. Illumination is the moment of breakthrough, the eureka. Verification is the assessment of whether the solution meets creative criteria.

Case-Based Reasoning uses past successful solutions to solve new problems. Typically, a system searches through all cases in memory, applies a similarity metric, and returns the case most similar to the new problem. Janet Kolodner argued that the most useful cases are those that address the reasoner's current goal, which may not always be the most similar cases, especially when the aim is creativity.

When cases are treated as collections of pieces rather than monolithic entities, systems can draw from multiple cases to construct solutions, allowing more flexible knowledge representation essential for creativity.

A plan is a sequence of steps or actions designed to achieve a goal. Case-Based Planning systems reuse sequences from past plans to build new ones. Systems such as CELIA, JULIA, and PRODIGY/ANALOGY break problems into smaller sub-goals, enabling plan construction through composition of sub-plans. This produces one of the most common structures for plan cases: hierarchical representation, resembling a tree where each node is a goal and its children are sub-goals, terminating in actions. Each goal depends on other goals, especially in structured domains.

Recent years have seen substantial research into computational creativity, with the Georgia Institute of Technology playing a significant role. Ashwin Ram and colleagues argue that creativity results from mechanisms continuous with those used in ordinary thinking. They view creativity as an extraordinary outcome achieved through ordinary mechanisms applied with conscious strategic control.

Linda Wills and Janet Kolodner proposed a model for creative design involving three steps: enumerating alternative solutions, re-description and elaboration of problem specifications, and evaluation of proposed solutions. In creative design, A. Goel, S. Bhatta, and S. Prabhakar developed IDEAL, which represents knowledge through behavior, functionality, and structure.

Ashwin Ram distinguished two types of creative processes: constructive conceptual change and extrapolative conceptual change. Kenneth Moorman and Ram proposed a model for creative understanding, separate from creative invention or creative problem solving, implemented in the ISAAC system for creative reading of science fiction.

Douglas Hofstadter and Melanie Mitchell worked on COPYCAT, a nondeterministic analogy-making system that models both human analogy-making and creative analogy.

Analogy-making is central to creativity. The systems mentioned use it to apply knowledge from a familiar problem, called the source analog, to solve a target problem that shares analogous features.

This article presents the INSPIRER model (ImagiNation taking as Source Past and Imperfectly RElated Reasonigs) for the Creative Planning Process. An implementation called SICOM applies this model to music composition. Divergent production of creative music pieces uses hierarchical representations assembled from pieces of past works. This is achieved either by applying a similarity metric with varied parameter values across problem-solving sessions, or by establishing different degrees of originality for different hierarchical levels in each session.

A Computational Model for the Creative Planning Process

Preparation

Preparation involves acquiring background knowledge and formulating the problem.

Background Knowledge

The creative process draws on previous knowledge, represented as cases. Because the system deals with creative planning, cases are plans. They are stored incrementally as they are experienced, following standard case-based learning.

A plan case is a hierarchically arranged set of goals and actions. The main goal is broken into sub-goals, and eventually into actions represented at the leaf nodes. Although actions appear as leaves, some of their attributes are inherited from ancestors higher in the hierarchy.

Each node in the hierarchy is a case piece. Links between pieces represent causal justifications or explanations. Hierarchical links show goal refinement; temporal links reflect causal relationships over time.

A case piece contains seven types of information: a unique name; the name of the case it belongs to; its address within the case; constraints; a set of attribute/value pairs; antecedents; and consequents.

The address indicates the piece's hierarchical level and temporal position within the case.

Constraints are also attribute/value pairs. A set of constraints C = {a1 = vc1, a2 = vc2, ..., an = vcn} for a piece p means that if p's ancestors have any of the attributes a1 through an, their values must match vc1 through vcn. Otherwise, the piece is incoherent with its ancestors, and constraints determine whether a piece can fill a missing piece in a solution.

Antecedents and consequents are causal links. Antecedents show which other pieces explain a given piece. Consequents show which pieces are explained by the given piece.

The context of a case piece is the set of surrounding case pieces. Eight types of contexts exist, depending on the kind of link between the piece and its neighbors.

Problem Formulation

A new problem given to the system consists of a set of linked case pieces. At minimum, the main goal (root case piece) must be provided with its name, address, constraints, and attributes filled in.

The meaning of a problem description with main goal, sub-goals, and actions is that the system must find a structured plan solution that achieves the goal, sub-goals, and performs the specified actions.

Incubation

During incubation, an entire solution is constructed by iteratively retrieving case pieces from memory to fill missing pieces in a tree-like partial solution. This proceeds level by level, from highest to lowest, and from leftmost to rightmost within each level.

The retrieval process for a missing piece follows these steps:

1. Determine the set of candidate case pieces that belong to the same hierarchical level as the missing spot.
2. Apply a constraint-based filter. A candidate piece p is kept if, for every constraint in p's set where the attribute matches an attribute of the missing piece's ancestors, the constraint value equals the ancestor's attribute value. If not, p is filtered out.
3. Calculate similarity using a metric that examines attributes, address, and context:

CasePieceSim(p, πi) = α × AttrSim(p, πi) + φ × AddrSim(p, πi) + λ × ContSim(p, πi)

Here, α, φ, and λ represent weights assigned by the user.

4. Rank the candidate pieces by similarity score.
5. Select a case piece.
6. Validate placement. This checks for link incompatibilities between the chosen piece and the partially constructed solution. Incompatibilities are resolved either by relaxing them or by choosing another piece.

Illumination

Illumination is the step where the solution constructed during incubation is proposed. However, the solution may still have missing pieces. Each unresolved gap initiates another round of preparation, incubation, and illumination, continuing until every missing piece is filled. That new solution may itself create more sub-problems.

Verification

The full solution undergoes a validation test for originality and appropriateness. In the current version of INSPIRER, the user performs this verification.

Music Composition Domain

The application uses analysis of music pieces from a seventeenth-century composer as a foundation for divergent production of new music pieces. Music can be represented as a hierarchy of temporal objects — objects with an associated duration. The system uses six hierarchical levels, sufficient for representing all relevant aspects of the music. Each piece is treated as a plan: a sequence of musical objects aiming to achieve a musical goal, such as a sonata.

Divergent Production of Music Pieces

Divergent production creates several music pieces, some potentially creative (original and appropriate) and others uncreative (neither original nor appropriate).

One method for divergent production in this model is to assign different values to the similarity metric parameters α, φ, and λ in different problem-solving sessions for the same problem. Another is to give these parameters different values for each hierarchical level, thereby seeking different degrees of originality at different levels. This makes sense for domains with hierarchical knowledge representation — for example, musical composition, screenplay writing, and general writing — where creative solutions often share similar high-level structures but differ markedly at lower levels.

Two main groups of planning sessions were conducted in the music domain: Session Group I and Session Group II. The goal was to produce different creative music solutions in each session.

Session Group I (#I1 through #I10) assigned varying weights to α (Attribute Similarity), φ (Address Similarity), and λ (Context Similarity).

Session Group II (#II1 through #II4) assigned different weights to the parameters at different hierarchical levels. For instance, Level 0 and Level 5 each received a parameter value of 6%, while Level 2 and Level 3 received 13%.

Problem proposed in each session was identical: generate a new musical melody, such as for a sonata. Only one past case solution (one melody) was stored in memory, to test performance under minimal knowledge—the worst conditions for producing original solutions. Each session generated one new case solution (a new melody), which was empirically evaluated for appropriateness and originality against the original composer’s piece, the latter assigned 100% appropriateness. In brief, originality is calculated as the number of new relations between case pieces in the new solution divided by the total possible new relations.

Session Group I includes ten sessions (Session #I1 through Session #I10), and Session Group II contains four sessions (Session #II1 through Session #II4).

Sessions in Group I reveal how adjusting the weight of each term in the similarity metric affects outcomes. In Session #I1, all three parameters—address, attribute, and context—were set to 1, giving them equal relevance (approximately 33% each). Session #I2 reduced the attribute term to 1/11 (about 9% relevance), while address and context each received 5/11 (roughly 45%). In Sessions #I3 and #I4, the weight of the attribute term dropped further: 1/21 (≈ 4.7%) and 1/31 (≈ 3.2%) respectively. Sessions #I5, #I6, and #I7 mirror #I2, #I3, and #I4, but now the context term is the least relevant. Similarly, in Sessions #I8, #I9, and #I10, the address term bears the smallest relevance.

Comparing Session #I1 with Sessions #I2–#I4, #I5–#I7, or #I8–#I10 shows that reducing the importance of a similarity metric term yields solutions with more incorrectly selected case pieces—and therefore higher originality (Figure 5). A case piece is considered incorrectly (or wrongly) selected when it differs from the piece that would have been chosen had the similarity metric operated with perfect accuracy. For example, the solution produced by Session #I3 is more original than that from Session #I2, which in turn is more original than Session #I1’s result.

Results in Figure 5 indicate that the address term is the most critical component of the similarity metric: when it is given the least weight (in Sessions #I8, #I9, and #I10), the number of incorrectly selected case pieces is highest. Session #I10, in fact, yielded the solution with the most wrongly selected case pieces and thus the greatest degree of originality.

Figure 5: Results of Session Group I.

Even though the degrees of originality differ substantially across sessions, appropriateness remains relatively stable (Figure 5). This is primarily due to the validation step in the construction process, which prevents incoherence.

Notably, the number of case pieces wrongly selected by the similarity metric generally exceeds the number mistakenly selected after the evaluation step (Figure 5). Again, the validation step is responsible: it rejects some pieces selected by the metric because they are inconsistent with the partial solution under construction. Furthermore, the number of incorrectly selected pieces after evaluation tends to be greater than the solution’s degree of originality. This occurs because an incorrectly selected case piece may retain some old relations from the remembered past case.

Sessions in Group II focus on specifying desired originality for different hierarchical levels of the solution. The global originality of the whole solution is the average of originality across all levels. In Session #II1, the desired originality for all six levels was 0%—meaning the solution should match the one stored in memory exactly. Session #II2 aimed for about 16% originality in levels 5, 4, and 3; around 13% in level 2; approximately 6% in level 1; and about 0% in level 0. Session #II3 mirrors #II2 but requests the same degrees of originality in reverse order: levels 0, 1, and 2 desire 16% originality, while levels 3, 4, and 5 aim for 13%, 6%, and 0% respectively.

Drawing from Session Group I, the parameter values βα, ∅, and λ from Sessions #I4, #I3, #I2, and #I1 yielded solutions with approximately 16%, 13%, 6%, and 0% originality. It therefore seems reasonable to apply, for instance, the parameters of Session #I4 in the similarity metric to construct levels 5, 4, and 3 in Session #II2, the parameters of Session #I3 for level 2, and so on (Figure 4).

Figure 6 shows that Session #II1 produced a solution with no originality and 100% appropriateness, as expected. Session #II2 yielded a slightly more original solution than Session #II3, even though the desired degrees of originality were similar. This is because Session #II3 assigned higher originality to the highest levels, where the number of case pieces is low and thus the number of possible recombinations is also low. Additionally, appropriateness in Session #II3 is marginally lower than in Session #II2, due to structural incoherences arising at the highest levels.

As anticipated, Session #II4 produced the most original solution but with the least appropriateness.

These findings suggest that Session #II2’s solution is the most creative: it achieves good appropriateness together with considerable originality.

As in Session Group I, and for the same reason, the number of case pieces wrongly selected by the similarity metric is greater than the number wrongly selected after the evaluation step, and this in turn exceeds the solution’s degree of originality.

5 Conclusions

We have presented an approach for constructing diverse creative music pieces, which we regard as plans. Each music piece is organized as a tree-like network where pieces are linked to one another.

By recombining case pieces retrieved from a single past successful case and applying a similarity metric with varying parameter values across sessions, a range of creative solutions is produced. This metric evaluates the similarity between two case pieces based on the similarity of their addresses, attributes, and contexts.

Acknowledgements

We thank the Coimbra School of Music for their valuable contribution.

References