Math and the Musical Offering

6. The transformation  g(t) = -f((t-0.5)/2) + L  and Canon 4

In Canon 4 (per Augmentationem contrario Motu) Voice 2 starts at the sign. The second clef is again upside-down, signalling that Voice 2 runs upside-down (as in Canon 3).

Bach's score for Canon 4

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

The second clef is a treble clef, and positions Voice 2 (which enters halfway through the first measure) above the modified Royal Theme in the top staff. The Royal Theme now appears as a middle voice. But there is more: the ``per augmentationem'' in the title indicates that in Voice 2 each note has double the value that it had in Voice 1, so Voice 2 moves with half the speed of Voice 1.

Function Example To make the function g copy the function f but move half as fast, we define g(t) = f(t/2) so that g(2) = f(1), g(4) = f(2), etc. To make g also start half a measure later and higher and move in the reverse direction, we combine the previous modifications and set

g(t) = -f((t-0.5)/2) + L.

As in Canon 3, the L is chosen to position our graph approximately where Voice 2 is positioned. Note that it takes 16 measures to hear the whole canon: Voice 1 has to play its tune twice before Voice 2 is finished.

© copyright 1999, American Mathematical Society.