Abstract:

Music education contributes to an individual’s wellbeing, stimulating creativity, and improving memory and communication, providing skills that can be transferred to other fields. Built using highly structured compositions patterns, music, and the study of it also provide a mechanism to bridge science and mathematics and creative arts. Musical analyses are very important in music education especially when discussing the styles of different composers. This is often done by identifying repetitive patterns in composition, also known as motifs. Traditional music education has students identify such motifs manually. Yet, current musical analysis techniques fail to look for other transformed versions of motifs.

Music is inherently related to mathematics and mathematical transformations of musical motifs have been found in many composers’ works. A musical motif can also be transformed geometrically within a composition: it can be repeated (translated horizontally), transposed (translated vertically), inverted (horizontally mirrored), in retrograde (vertically mirrored), in retrograde inversion (rotated by 180°) and even show up in these combinations but with different note values (dilated or contracted).

In this paper we first discuss how analysis of music patterns can be done more efficiently by identifying mathematical transformations of the motifs (such as transpositions or mutations). Geometrically transformed motifs are especially interesting due to their unique structures and the fact that they could retain a musical piece’s general melody while evading copyright detection efforts. Identifying the motifs and their transformations can also be helpful to composers attempting to replicate a certain style by creating original motifs and melodies on the transformational structure of another composer. Next, we present an Automatic Algorithmic Transformed Motif Finder (AATMoF) that streamlines motif discovery using music encoded in MIDI files and its implementation in a computer application. We test AATMoF on two classical compositions by Johann Sebastian Bach and compare its performance to a human analysis of both pieces. Bach is an especially prolific user of motifs and often has complex structures within his music. The two pieces were selected due to their differences and contrasting complexity in the use of geometrical transformations of the main motif. We then discuss how the application is an integral part of a larger project that focuses on developing specific composer-like compositions using mathematically transformations of original motifs as a way to build students’ understanding of both mathematics and music.

Keywords:

Blending of music and mathematics education, motifs, musical analysis, mathematical transformations.