Symmetries In Classical Music. From Guillaume de Machaut to Cesar Franck

Abstract

Reflection, translation, rotation are terms frequently used in mathematics and generally associated with the sense of sight: we can use tiles or stained glass to clarify these concepts, but it is possible to exemplify the same concepts using the sense of hearing. In the first case the x and y coordinates of the plane are, for example, the width and height of a stained-glass window, in the second case the variables are respectively time and sound height, but instead of vision, we need hearing and memory to recognize the symmetry. Musicians and composers are aware of these con- cepts and use them with rigor and ingenuity. We present several musical examples from different eras, analysing compositions ranging from Guillaume de Machaut (14th century) to Cesar Franck (19th century), through Mozart's joviality and Bach's genius (18th century).