Math and the Musical Offering

For background on Bach and his work, including the story of the Musical Offering and why the ``royal theme'' is so called, see Tim Smith's Web site The Canons and Fuguesof J. S. Bach.

Johann Sebastian Bach's Musical Offering contains ten canons. In each of these canons a musical line is played twice (or four times in Canon 10). The second version is always transformed with respect to the first by shifting in time, but it may also be shifted in pitch, turned upside-down, stretched, or played backwards. Each of these transformations occurs in the mathematics of elementary functions; they are examples of how new functions can be made out of old and of how a function can be tailored to fit a new situation. We will look at some simple transformations and see how they are exemplified in the first five of the Musical Offering canons.

• What is a canon?

   

• g(t) = f(t-1) and Canon 2

   

• g(t) = f(t-1) + H and Canon 5

   

• g(t) = -f(t-0.5) + K and Canon 3

   

• g(t) = -f((t-0.5)/2) + L and Canon 4

   

• g(t) = f(18-t) and Canon 1

   

Food for thought: There is one elementary transformation of functions that does not appear in any of Bach's canons. Which is it and why?

--Tony Phillips