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.

Bach's score for Canon 3

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.

Function Example 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.