Computational Music Analysis

This chapter examines questions of what is to be analysed in computational music analysis, what is to be produced, and how one can have confidence in the results. These are not new issues for music analysis, but their consequences are here considered explicitly from the perspective of computational analysis. Music analysis without computers is able to operate with multiple or even indistinct conceptions of the material to be analysed because it can use multiple references whose meanings shift from context to context. Computational analysis, by contrast, must operate with definite inputs and produce definite outputs. Computational analysts must therefore face the issues of error and approximation explicitly. While computational analysis must retain contact with music analysis as it is generally practised, I argue that the most promising approach for the development of computational analysis is not systems to mimic human analysis, but instead systems to answer specific music-analytical questions. The chapter concludes with several consequent recommendations for future directions in computational music analysis.

Selecting an appropriate representation for chords is important for encoding pertinent harmonic aspects of the musical surface, and, at the same time, is crucial for building effective computational models for music analysis. This chapter, initially, addresses musicological, perceptual and computational aspects of the harmonic musical surface. Then, two novel general chord representations are presented: the first, the General Chord Type (GCT) representation, is inspired by the standard Roman numeral chord type labelling, but is more general and flexible so as to be applicable to any idiom; the second, the Directed Interval Class (DIC) vector, captures the intervallic content of a transition between two chords in a transposition-invariant idiom-independent manner. Musical examples and preliminary evaluations of both encoding schemes are given, illustrating their potential to form a basis for harmonic processing in the domain of computational musicology.

We propose a spatial approach to musical analysis based on the notion of a chord complex. A chord complex is a labelled simplicial complex which represents a set of chords. The dimension of the elements of the complex and their neighbourhood relationships highlight the size of the chords and their intersections. Following a well-established tradition in set-theoretical and neo-Riemannian music analysis, we present the family of T/I complexes which represent classes of chords which are transpositionally and inversionally equivalent and which relate to the notion of Generalized Tonnetze. A musical piece is represented by a trajectory within a given chord complex. We propose a method to compute the compactness of a trajectory in any chord complex. Calculating the trajectory compactness of a piece in T/I complexes provides valuable information for music analysis and classification. We introduce different geometrical transformations on trajectories that correspond to different musical transformations. Finally, we present HexaChord, a software package dedicated to computer-aided music analysis with chord complexes, which implements most of the concepts discussed in this chapter.

In this chapter, we review and elaborate a methodology for contextual multi-scale set-class analysis of pieces of music. The proposed method provides a systematic approach to segmentation, description and representation in the analysis of the musical surface. The introduction of a set-class description domain provides a systematic, mid-level, and standard analytical lexicon, which allows for the description of any notated music based on a fixed temperament. The method benefits from representation completeness, a balance between generalization and discrimination of the set-class spaces, and access to hierarchical inclusion relations over time. Three new data structures are derived from the method: class-scapes, class-matrices and class-vectors. A class-scape represents, in a visual way, the set-class content of each possible segment in a piece of music. The class-matrix represents the presence of each possible set class over time, and is invariant to time scale and to several transformations of analytical interest. The class-vector summarizes a piece by quantifying the temporal presence of each possible set class. The balance between dimensionality and informativeness provided by these descriptors is discussed in relation to standard content-based tonal descriptors and music information retrieval applications. The interfacing possibilities of the method are also discussed.

Can a computer understand musical forms? Musical forms describe how a piece of music is structured. They explain how the sections work together through repetition, contrast, and variation: repetition brings unity, and variation brings interest. Learning how to hear, to analyse, to play, or even to write music in various forms is part of music education. In this chapter, we briefly review some theories of musical form, and discuss the challenges of computational analysis of musical form. We discuss two sets of problems, segmentation and form analysis. We present studies in music information retrieval (MIR) related to both problems. Thinking about codification and automatic analysis of musical forms will help the development of better MIR algorithms.

Voice separation is the process of assigning notes to musical voices. A fundamental question when applying machine learning to this task is the architecture of the learning model. Most existing approaches make decisions in note-to-note steps (N2N) and use heuristics to resolve conflicts arising in the process. We present here a new approach of processing in chord-to-chord steps (C2C), where a solution for a complete chord is calculated. The C2C approach has the advantage of being cognitively more plausible but it leads to feature modelling problems, while the N2N approach is computationally more efficient. We evaluate a new C2C model in comparison to an N2N model using all 19 four-voice fugues from J. S. Bach’s Well-Tempered Clavier. The overall accuracy for the C2C model turned out slightly higher but without statistical significance in our experiment. From a musical as well as a perceptual and cognitive perspective, this result indicates that feature design that makes use of the additional information available in the C2C approach is a worthwhile topic for further research.

Recent developments in computational linguistics offer ways to approach the analysis of musical structure by inducing probabilistic models (in the form of grammars) over a corpus of music. These can produce idiomatic sentences from a probabilistic model of the musical language and thus offer explanations of the musical structures they model. This chapter surveys historical and current work in musical analysis using grammars, based on computational linguistic approaches. We outline the theory of probabilistic grammars and illustrate their implementation in Prolog using PRISM. Our experiments on learning the probabilities for simple grammars from pitch sequences in two kinds of symbolic musical corpora are summarized. The results support our claim that probabilistic grammars are a promising framework for computational music analysis, but also indicate that further work is required to establish their superiority over Markov models.

In a harmonic analysis task, melodic analysis determines the importance and role of each note in a particular harmonic context. Thus, a note is classified as a harmonic tone when it belongs to the underlying chord, and as a non-harmonic tone otherwise, with a number of categories in this latter case. Automatic systems for fully solving this task without errors are still far from being available, so it must be assumed that, in a practical scenario in which the melodic analysis is the system’s final output, the human expert must make corrections to the output in order to achieve the final result. Interactive systems allow for turning the user into a source of high-quality and high-confidence ground-truth data, so online machine learning and interactive pattern recognition provide tools that have proven to be very convenient in this context. Experimental evidence will be presented showing that this seems to be a suitable way to approach melodic analysis.

We describe and discuss our computer implementations of Lerdahl and Jackendoff’s (1983) Generative Theory of Tonal Music (GTTM). We consider this theory to be one of the most relevant music theories with regard to formalization because it captures aspects of musical phenomena based on the Gestalts perceived in music and presents these aspects with relatively rigid rules. However, the theory has several problems in terms of computer implementation. To overcome these problems, we have proposed four different kinds of analyser: an automatic timespan tree analyser (ATTA); a fully automatic time-span tree analyser (FATTA); the sGTTM analyser, which detects local grouping boundaries by combining GTTM with statistical learning using a decision tree; and the sGTTM-II analyser, with which we introduce full parameterization and statistical learning.

In this chapter, we present an algebraic framework in which a set of simple, intuitive operations applicable to music can be flexibly combined to realize a target application and generate music. We formalize the concept of time-span tree introduced by Lerdahl and Jackendoff (1983) in their Generative Theory of Tonal Music (GTTM) and define the distance between time-span trees, on the hypothesis that this might coincide with the psychological resemblance between melodies heard by human listeners. To confirm the feasibility of the proposed framework, we conduct an experiment to determine whether the distance calculated on the basis of the framework reflects cognitive distance in human listeners. To demonstrate that the algebraic framework is computationally tractable, we present the implementation of a musical morphing system that, given two original melodies, generates an intermediate melody at any internally dividing point between them (i.e., at any ratio).

Motivic analysis provides very detailed understanding of musical compositions, but is also particularly difficult to formalize and systematize. A computational automation of the discovery of motivic patterns cannot be reduced to a mere extraction of all possible sequences of descriptions. The systematic approach inexorably leads to a proliferation of redundant structures that needs to be addressed properly. Global filtering techniques cause a drastic elimination of interesting structures that damages the quality of the analysis. On the other hand, a selection of closed patterns allows for lossless compression. The structural complexity resulting from successive repetitions of patterns can be controlled through a simple modelling of cycles. Generally, motivic patterns cannot always be defined solely as sequences of descriptions in a fixed set of dimensions: throughout the descriptions of the successive notes and intervals, various sets of musical parameters may be invoked. In this chapter, a method is presented that allows for these heterogeneous patterns to be discovered. Motivic repetition with local ornamentation is detected by reconstructing, on top of “surface-level” monodic voices, longer-term relations between non-adjacent notes related to deeper structures, and by tracking motives on the resulting syntagmatic network. These principles are integrated into a computational framework, the MiningSuite, developed in Matlab.

Several point-set pattern-discovery and compression algorithms designed for analysing music are reviewed and evaluated. Each algorithm takes as input a point-set representation of a score in which each note is represented as a point in pitch-time space. Each algorithm computes the maximal translatable patterns (MTPs) in this input and the translational equivalence classes (TECs) of these MTPs, where each TEC contains all the occurrences of a given MTP. Each TEC is encoded as a 〈pattern, vector set〉 pair, in which the vector set gives all the vectors by which the pattern can be translated in pitch-time space to give other patterns in the input dataset. Encoding TECs in this way leads, in general, to compression, since each occurrence of a pattern within a TEC (apart from one) is encoded by a single vector, that has the same information content as one point. The algorithms reviewed here adopt different strategies aimed at selecting a set of MTP TECs that collectively cover (or almost cover) the input dataset in a way that maximizes compression. The algorithms are evaluated on two musicological tasks: classifying folk song melodies into tune families and discovering repeated themes and sections in pieces of classical music. On the first task, the best-performing algorithms achieved success rates of around 84%. In the second task, the best algorithms achieved mean F1 scores of around 0.49, with scores for individual pieces rising as high as 0.71.

The task of recognizing a composer by listening to a musical piece used to be reserved for experts in music theory. The problems we address here are, first, that of constructing an automatic system that is able to distinguish between music written by different composers; and, second, identifying the musical properties that are important for this task. We take a data-driven approach by scanning a large database of existing music and develop five types of classification model that can accurately discriminate between three composers (Bach, Haydn and Beethoven). More comprehensible models, such as decision trees and rulesets, are built, as well as black-box models such as support vector machines. Models of the first type offer important insights into the differences between composer styles, while those of the second type provide a performance benchmark.

Comparing groups in data is a common theme in corpus-level music analysis and in exploratory data mining. Contrast patterns describe significant differences between groups. This chapter introduces the task and techniques of contrast pattern mining and reviews work in quantitative and computational folk music analysis as mining for contrast patterns. Three case studies are presented in detail to illustrate different pattern representations, datasets and groupings of folk music corpora, and pattern mining methods: subgroup discovery of global feature patterns in European folk music, emerging pattern mining of sequential patterns in Cretan folk tunes, and association rule mining of positive and negative patterns in Basque folk music. While this chapter focuses on examples in folk music analysis, the concept of contrast patterns offers opportunities for computational music analysis more generally, which can draw on both musicological traditions of quantitative comparative analysis and research in contrast data mining.

Pattern discovery is an essential computational music analysis method for revealing intra-opus repetition and inter-opus recurrence. This chapter applies pattern discovery to a corpus of songs for the bagana, a large lyre played in Ethiopia. An important and unique aspect of this repertoire is that frequent and rare motifs have been explicitly identified and used by a master bagana teacher in Ethiopia. A new theorem for pruning of statistically under-represented patterns from the search space is used within an efficient pattern discovery algorithm. The results of the chapter show that over- and under-represented patterns can be discovered in a corpus of bagana songs, and that the method can reveal with high significance the known bagana motifs of interest.

Did Ludwig van Beethoven (1770–1827) re-use material when composing his piano sonatas? What repeated patterns are distinctive of Beethoven’s piano sonatas compared, say, to those of Frédéric Chopin (1810–1849)? Traditionally, in preparation for essays on topics such as these, music analysts have undertaken inter-opus pattern discovery—informally or systematically—which is the task of identifying two or more related note collections (or phenomena derived from those collections, such as chord sequences) that occur in at least two different movements or pieces of music. More recently, computational methods have emerged for tackling the inter-opus pattern discovery task, but often they make simplifying and problematic assumptions about the nature of music. Thus a gulf exists between the flexibility music analysts employ when considering two note collections to be related, and what algorithmic methods can achieve. By unifying contributions from the two main approaches to computational pattern discovery—viewpoints and the geometric method—via the technique of symbolic fingerprinting, the current chapter seeks to reduce this gulf. Results from six experiments are summarized that investigate questions related to borrowing, resemblance, and distinctiveness across 21 Beethoven piano sonata movements. Among these results, we found 2–3 bars of material that occurred across two sonatas, an andante theme that appears varied in an imitative minuet, patterns with leaps that are distinctive of Beethoven compared to Chopin, and two potentially new examples of what Meyer and Gjerdingen call schemata. The chapter does not solve the problem of inter-opus pattern discovery, but it can act as a platform for research that will further reduce the gap between what music informaticians do, and what musicologists find interesting.