Thomas M. J. Schäfer composed a permutative canon for 3 voices, 30 bars and 300 seconds. The graphical score has the shape of an equilateral triangle. The canon theme consists of 10 bars. The three voices have to pause sometimes for one or more bars when the theme was played. The beginning tonality is shifted, the permutations lead to dissonances and modern impression. Baroque and modern spirit are linked by romantic phrasing. The canon refers to the name, the oeuvre and the contrapuntal execution of J. S. Bach.

This video was part of the exhibition “Geometry and Imagination: Patterns in Nature and Culture” presented during ICM 2018.

J. S. Bach probably is one of the composers with most affinity to mathematics. He developed the art of fugue in a programmatic way in his work “The Art of Fugue”. So he used all kinds of techniques that are characteristic for composing fugues and canons. Canons are a special case of fugues. In the most strictest form they consist of just one melody that starts to different times in at least two voices. The canon melody repeats several times in a loop, at least one time for each voice. There are many ways to vary the melody with transformations that are well known from geometry: translation (transposition), rotation, scaling, mirror transformations. Bach liked composing canons and fugues so much that he used this technique even implicit. His great mass in B Minor ends with the Dona nobis pacem, where a canon melody is varied and hard to discover. The animation shows how the canon is constructed and a little excerpt of the beginning and the end of that movement.