Mathematical characterization of music-theoretic structures using Haskell

Abstract

Music theory enables an individual to understand how different notes and rhythms are combined to create some composition. It also enables a musician to predict what note might be played next and to understand what has changed if this prediction fails. To establish such predictions musicians use music-theoretic structures to identify how different sounds relate to each other and what this relation implies. Using different geometrical properties or a geometrical algorithm known as Filtered Point-Symmetry, one can obtain the bulk of these music-theoretic structures and geometrically represent them. However, this process may be tedious and sometimes hard to compute, given the large number of calculations required. This project presents an application that enables a user to either input the parameters of the previously mentioned algorithm and compute the structure, or enable the user to request pitch class sets with specific attributes and properties. Since geometry requires algebraic representation and computation, it was deemed appropiate that a purely functional language, such as Haskell was used for these computations. To obtain pitch class sets with specifi c properties, the algorithm is required to generate all possible pitch class sets and then identify which sets maximise the requested property. On the other hand, the Filtered Point-Symmetry algorithm involves a number of geometrical operations. A user interface was then built using C++. This GUI provides the user the ability to easily communicate with the back-end. Once the result(s) are obtained, the front-end graphically renders the structure. If more than one structure is obtained, the user may choose which structure to render. The Filtered-Point Symmetry Algorithm successfully reproduced all results from [DHJ08]. The Maximisation algorithm was tested on twelve and nineteen tone equal temperament systems (TET), since these are the most researched systems. All twelve TET pitch class sets presented by Rappaport in [Rap06] were successfully reproduced. Regarding nineteen tonality, all results presented by Mandelbaum, who pioneered this system in [Man61] were also obtained.