## Math and the *Musical Offering*

## 5. The transformation
*g*(*t*) = -*f*(*t*-0.5) + *K* and Canon 3

**Canon 3** is described by Bach as *a 2 per Motum contrarium*.

Graphic © 1996,
Timothy A. Smith, used by permission.

Here the upside-down second signature in the lower staff indicates that
the second voice is to be played upside-down. Voice 1 starts on C, and
Voice 2 starts on G (a perfect fourth lower) at the sign and moves in
the opposite direction from the first. Again, a modified Royal Theme
plays in the top staff, in harmony with the two lower voices. Note that
here Bach scored the canon in 4 measures and had the second voice start
halfway through the first measure.

To make a function run upside-down only requires a minus sign. In this
example the blue function *g* is defined from the red function *f*
by

*g*(*t*) = -*f*(*t*-0.5) + *K*.
Since there is no natural meaning for 0 pitch, there is no natural
starting point on the vertical axis; on the other hand using a minus sign
implies knowing where 0 is. The practical consequence is that the
vertical translation part *K* depends on the (artificial) location
of 0. Here it was chosen to put the red and blue starting points
in approximately correct relative position.

© copyright 1999, American Mathematical Society.