Science Magazine: The Geometry of Musical Chords

The Geometry of Musical Chords - Dmitri Tymoczko

This magazine is a great source in establishing the history of how geometry got interlinked with the making of music. We are introduced to the music aspect of the association through the definition of what exactly a musical chord is. "A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries and suggest different musical uses. Western music lies at the intersection of two seemingly independent disciplines: harmony and counterpoint. Harmony delimits the acceptable chords (simultaneously occurring notes) and chord sequences. Counterpoint (or voice leading) is the technique of connecting the individual notes in a series of chords so as to form simultaneous melodies. Chords are usually connected so that these lines (or voices) move independently (not all in the same direction by the same amount), efficiently (by short distances), and without voice crossings (along non intersecting paths). These features facilitate musical performance, engage explicit aesthetic norms, and enable listeners to distinguish multiple simultaneous melodies. The preceding ideas can be extended in several directions. First, one might examine in detail how composers have exploited the geometry of musical chords. Second, one could generalize the geometrical approach by considering quotient spaces that identify transpositionally and inversionally related chords. Third, because cyclical rhythmic patterns can also be modeled as points, one could use these spaces to study African and other non-Western rhythms. Fourth, one could investigate how distances in the orbifolds relate to perceptual judgments of chord similarity. Finally, understanding the relation between harmony and counterpoint may suggest new techniques to contemporary composers."

The information gathered in this magazine uses terms that both music and math students would be familiar with. I had to read this article several times to get an understanding of the links between geometry and musical chords, but I thought it was intriguing how simple the techniques seem when they are being used for a particular purpose. I'm sure I am qualified to review such an informative piece but I thought it was quite interesting because it really opens up the mind to make new associations with a topic such as geometry which has been introduced to us in our earlier years. Also , probably knowing the extent of such how geometry is used would help us put a bigger emphasis on its importance.

Reference:

Science 7 July 2006:Vol. 313. no. 5783, pp. 72 - 74DOI: 10.1126/science.1126287