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.

--*Tony Phillips*

- Introduction
- 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

© copyright 1999, American Mathematical Society.