Functional Harmonic Analysis Using Probabilistic Models

A variety of musical analysis techniques, often collectively referred to as functional harmonic analysis, represents a musical passage as a sequence of chords. The chords are expressed in terms of their function (e.g., dominant or tonic), often written with corresponding Roman numerals like V or I. Each chord is analyzed in the context of a key, which might modulate over time. This article focuses on the study of algorithms for this type of analysis.

An obvious application of algorithmic harmonic analysis would be locating musical examples in a database matching a particular harmonic query, for example, "What are the earliest examples of the use of German augmented sixth chords or Neapolitan chords?" "Which Beatles songs have deceptive cadences?" "Where can I buy the piece I heard on the radio with the harmonic progression I vi IV V I repeated many times?" It is likely that such applications will be most useful to musicologists, because the mere formulation of such queries requires a more sophisticated understanding of harmony than would be expected of an average music enthusiast.

Another application of this type of harmonic analysis could be found in score-writing programs. When these programs produce musical notation from MIDI files, they often produce incorrect spellings of accidentals. Many of these errors could be avoided if the harmonic content were understood at a level beyond the nominal key signature, as in Chew and Chen (2003).

A more subtle, yet perhaps more important, application might be one of representation. Harmonic analysis reduces music to a one-dimensional sequence of symbols from a small alphabet. The one-dimensional nature of this representation lends it to the wealth of search techniques treating strings as the basic unit of study. Such string-matching algorithms can find the string in a database minimizing a variety of edit-like distances in linear time. Pickens et al. (2002) show an example of a technique somewhat like harmonic analysis for representation and retrieval.

More generally, the one-dimensional musical reduction afforded by harmonic analysis might form the basis for genre classification or the construction of various music similarity metrics. Perhaps such analysis might even serve as a useful compositional tool by making unexpected links between musical passages, as in Peter Schickele's comical musical pastiches.

While we are interested in these applications, we find the study of the cognitive, or "artificial intelligence," aspect of harmonic analysis ample motivation by itself. Our basic approach is statistical; this orientation and methodology distinguishes our work from most other efforts we know. The most significant benefit of the probabilistic modeling we employ is the ability to learn aspects of our model in an unsupervised manner, for instance, using generic (unmarked) MIDI data. However, we also inherit computational machinery that identifies the best harmonic parse globally. In addition, we prefer the transparency and honesty of a clearly specified probabilistic model.

That said, we find common ground with several other previous efforts in harmonic analysis. Krumhansl (1990) identifies key by matching a histogram of pitches to a collection of possible key templates. Although our approach simultaneously identifies chord and key, the actual computation that measures the appropriateness of a particular key hypothesis is similar to that of Krumhansl. We share with Temperley and Sleator (1999) the recognition that rhythmic content is useful in harmonic analysis and the notion that harmonic analyses that fluctuate rapidly between keys are implausible and should be discouraged or penalized. Pardo (2002) builds upon this approach with a dynamic-programming algorithm optimized over the exponential number of segmentations in a computationally efficient manner. Similarly, dynamic programming is also fundamental to our work.

An overview of the approaches to algorithmic harmonic analysis is presented by Barthelemy and Bonardi (2001). Most approaches similar in scope to ours are rule-based: the music is reduced and recognized through a series of deterministic state transformations (merges, simplifications, intermediate labelings, etc.) moving systematically toward a final representation. In our view, there are two principal disadvantages of rule-based approaches. First, such schemes fail to articulate any measure of "goodness" of the possible "answers" and hence do not formulate the problem clearly...