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 website The Canons and Fugues of J. S. Bach.
Johan Sebastian Bach's Musical Offering contains 10 canons. In each of these canons a musical line is played twice (or 4 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.