Recognition of Variations Using Automatic Schenkerian Reduction

Experiments on techniques to automatically recognise whether or not an extract of music is a variation of a given theme are reported, using a test corpus derived from ten of Mozart‘s sets of variations for piano. Methods which examine the notes of the ‗surface‘ are compared with methods which make use of an automatically derived quasi-Schenkerian reduction of the theme and the extract in question. The maximum average F-measure achieved was 0.87. Unexpectedly, this was for a method of matching based on the surface alone, and in general the results for matches based on the surface were marginally better than those based on reduction, though the small number of possible test queries means that this result cannot be regarded as conclusive. Other inferences on which factors seem to be important in recognising variations are discussed. Possibilities for improved recognition of matching using reduction are outlined. 1. SCHENKERIAN REDUCTION Earlier work [6] has shown that Schenkerian analysis by computer is possible, though not easy. (Currently only short segments of music can be analysed, and confidence in the analyses produced cannot be high.) The aim of the research reported here is a first attempt at testing whether these automatic analyses produce information which is useful for information retrieval. Schenkerian analysis is a technique, with a long pedigree in music theory, which aims to discover the structural ‗framework‘ which is believed to underlie the ‗surface‘ of a piece of music (see [1], for example). Reduction according to the theory of Lerdahl & Jackendoff, which has also been subject to computational implementation [2], is broadly similar. Figure 1 shows the first four bars of the theme of a set of variations for piano by Mozart, and its reduction as derived by the software used here. (This is by far the simplest of the themes used here; to show other themes and their reductions would take more space than is available.) Schenker‘s reductions were notated in a different fashion, and also included information not given here, but the basic information of which pitches are regarded as more ‗structural‘, and so included in the higher levels, is similar. The research reported here fits into that body of MIR research which aims to improve MIR procedures through the application of ideas from music theory. 2. VARIATIONS A common type of composition in classical music is ‗theme and variations‘. In this kind of piece, a theme is presented, followed by a number of variations of that theme. There is no single and established definition of what constitutes a variation of a theme, but in the Classical period (the period of Haydn, Mozart and Beethoven) it is clear that a variation is not simply the presentation of the same melody in different arrangements (as it was for some later composers) but rather a composition which has the same structural features as the theme. This is particularly clear in Mozart‘s variations: they are almost always the same length as the theme, have the same number of phrases, and have matching cadences for those phrases (at least in their harmony; often in other features also). The