The geometry of melodic structure: Is there a strange attractor at the heart of Bach’s, Two Part Invention No. IV?

Abstract

Although certain works of music show melody lines that contain highly developed pitch organization, no study to date has demonstrated that for certain works this pitch organization may be chaotic in nature. In this paper, the principle melodic line for the TwoPartInvention, No. IV (for clavier) of Johannes Sebastian Bach has been analyzed. Taking a decidedly geometric attack, the pitch organization of the melodic material has been treated as a time series. In this particular example of counterpoint, the melodic line weaves continuously back and forth between the two lines of the clavier part. The melodic information has been converted into its associated pitch class numbers, mod 12. By concatenating the resulting numerical information of the two melody lines into one continuous line and treating it as a time series in pitch, it was possible to plot the three-dimensional phase space of the derived time series by use of time-delay plots. The attractor generated by this means, in this special case of musical counterpoint, indeed showed the requisite properties of a strange attractor. In addition to calculating the Lyopanov exponent, dimensional constants, and Fourier spectrum of this attractor, the following fascinating question was discussed: why is this data chaotic and how does it relate to the geometric structure of musical form in general?