A paradigmatic approach to extract the melodic structure of a musical piece

Abstract

We present an automated, mathematical approach to the paradigmatic analysis of the melodic content of a piece of music. We consider all melodic segments of consecutive notes, however, segments of different sizes are processed separately. We compare and group these segments using a similarity measure, which accounts for standard symmetry transformations such as translation and inversion. We then define a significance measure for melodies via the number of repeats of a given melody and its close variations in the piece, and extract the melodies which appear more often than a threshold value. These melodies are then clustered, and – for every cluster – the melody which is repeated most often (including repeats of its close variations) is selected as the cluster's representative. After identifying the paradigmatic elements of each piece, we analyse them using the representative melodies found by the new method. In the present paper, we use a terminology inspired by topology, and indicate related links to it. We test our approach on the Two Part Inventions of Johann Sebastian Bach. We find that the representative melodies identified by our approach agree with the results of the traditional music theory well. Additionally, because the analysis is restricted to segments of consecutive notes, the implementation is fast and results can usually be analysed without the need for elaborate post processing.

Acknowledgement

The authors would like to thank Gülen Ada TANIR for her kind support in the music-theoretical and analytical discussion of computational results.