Hidden Geometry in Music of Bach and Schoenberg: Reflection, Rotation, Proportion

Johnathan Saggau
Meeting Alhambra, ISAMA-BRIDGES Conference Proceedings (2003)
Pages 283–290


Musicians often equate the composers Bach (1685-1750) and Schoenberg (1874-1951) with a nearly obsessive relationship to numbers and mathematics. We think often of Schoenberg as a father of the numerical music organization systems of atonality and dodecaphony while there is an oft cited encoding within Bach's works the numerical and musical representation of his surname. Analysts often limit these composers' creativity to numerical surface details in the case of Bach or the tabulations of 12-tone row forms in the case of Schoenberg; this paper illuminates elegant architectural structures so prevalent behind such musical edifices. It comprises an architectural, geometrical, and statistical walking tour of two beautiful constructions, namely Schoenberg's Op. 19 No.2 for piano and the Courante from Bach's Suite for Solo Cello No.2. Through graphic measurements taken through modeling each piece of music onto a pitch-time two-dimensional complex plane (after Cogan and Escot), the paper analyzes internal architectural and geometric proportions of these works illuminating a consistent use of arithmetic, geometric, harmonic, and golden mean proportions amid these composers' works. It also provides graphical illumination of various pitch and time bilaterally symmetrical structures within the Schoenberg. Statistical contour correlations and oppositions are also found between ordered pitch data sets obtained from equal-length sections of the Bach using Spearman, Pearson, and Kendall data correlation methods. Finally, the paper compares the statistical distribution of pitch classes within the Schoenberg underlining his use of statistically lowest total duration pitches as contextually unique information at architecturally significant moments.