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Abstract

In this paper we explore the extent to which basic properties of music can be inferred by computers using machine learning techniques. Such patterns are determined rather easily to a trained human eye, but writing a function that maps notes or chords to the underlying "key center" is not trivial. For this reason we look to the power of multi-layer neural networks to explore this problem domain, feeding them increasingly preprocessed data to determine at what point the machine learning can fill in the remaining gaps. Training and test data are drawn from Beethoven piano sonatas, during which is represented a turning point in the way humans analyze key centers. As benchmarks we use not only raw scores, but comparisons with computer programs constructed from music-theoretic principles themselves (i.e., that do not require training sets).

Historical Background

The traditional art-music of Western Europe has the dubious fortune of centuries of scholarly analysis. Music theory has often been rightly accused of moving the domain of criticism away from audiences into rarefied academic circles, but its principles are what allow the creation of the complex works that both lay and learned enjoy. Knowledge of harmony in particular has evolved immensely in the last two centuries, and every discussion of it must begin with the concept of key center.

The idea that every note in a piece or section of a piece is related to a central note -- the key center -- is the abstraction that defines "tonal music." It is perhaps most easily recognized in indigenous music as the chord strummed underneath a soloist or the continuous drone of bagpipes. Cultures around the world typically add to this center a sequence of 5 or more notes called a scale, from which their melodies are constructed. For styles such as chant and plainsong, endless variation of simple scales encapsulates their music, or at least their harmony.

As Western music progressed into the early millenium, musicians quickly discovered that when overlaying different such melodies, harmonies could be produced beyond the key center of the scale being used. Thus, even when so-called polyphony went out of vogue in the 18th century, composers had acquired a wealth of experience manipulating harmonies, grouping them and the progressions between them into a framework beyond the scope of this paper. Nevertheless, around the same time they began considering the key center of their works important enough to add to the title. And so as the 19th century dawned lay the question: how many harmonies could a piece explore while still calling itself a Symphony in D?

Beethoven came of age at this cusp. His training was in the rigidly Classical form of Haydn and Mozart, but before his career was over he would almost singlehandedly inspire the Romantic age that pushed the very limits of tonal music. His piano sonatas may never be as famous as the 9th Symphony, but two of the most influential modern musicologists took special interest in them: Schenker edited them by hand, and Sch?nberg based his theory of "sentences" on their direct example; to every observer they represent a clear progression between eras. Our analysis of the pieces themselves will be minimal by comparison, but the observations we make tangential to our machine-learning discoveries may reflect this rich history.


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