A Link to the Past: An Examination of J.S. Bach and György Kurtág’s compositions using the golden ratio and Fibonacci Sequence

Abstract

Music, by nature, contains many connections with mathematics. Many scientists, musicians and thinkers beginning with Pythagoras in 6th century BC have explored these connections. In this presentation, using mathematical correlations, parallels are drawn between composers as disparate as Johann Sebastian Bach (1685-1750) and contemporary Hungarian composer György Kurtág (b. 1926). Through examination of the selected pieces from Bach’s Well Tempered Clavier and Kurtág’s Jatekok, the Fibonacci Series and the Golden Ratio may both be applied in various ways. The Golden Ratio appears in two of Bach’s preludes from Well Tempered Clavier, No. 1 in C major and No. 3 in C Sharp major, and serves to dictate the form of the pieces on a macro-level. Specifically, in the latter prelude, Bach creates a textural and rhythmic climax on the measure corresponding to the Golden Ratio. This is achieved through harmonic tension with a dominant chord as well as a break in the constant sixteenth note motion of the composition. By comparison, Kurtág’s laconic musical language requires a more specific micro-level examination. The Fibonacci Series may be applied to numerous miniatures in Jatekok through intervallic content, rhythm, and structure. For example, in many of the microludes examined, Kurtág chooses the intervals of minor second, major second, minor third, and perfect fourth. These intervals consist of one, two, three, and five half steps - all Fibonacci numbers. As with Bach’s music, in Kurtág’s Jatekok, the Golden Ratio is also present and is correspondent with the organization of notes and apex of the compositions. This occurs in Hommage a Kadosa through the composer’s choice of notes: of nine total notes in the piece, the one that occurs most often is C natural. The significance of this note is strengthened by not only its placement in the beginning and the end of the composition, but also its occurrence at the point of Golden Ratio. Utilizing these mathematical connections in music leads to a deeper understanding of the structure and semantics of a composition and also serves as a bridge between otherwise dissimilar composers.