Math in the Musical Offering
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.
Bach scored this canon in four measures. The upside-down second signature in the lower staff indicates thatthe second voice is to be played upside-down. Voice 1 starts on C; Voice 2 starts on G (a perfect fourth lower) at the sign (one-half measurelater) and moves in the opposite direction from Voice 1. Again, a modified ``royal theme''plays in the top staff in harmony with the two lower voices.
To make a function run upside-down requires only a minus sign. In thisexample the blue function g is defined from the red function fby g(t) = -f(t-0.5) + K.
Since there is no natural meaning for 0 pitch, there is no naturalstarting point on the vertical axis; on the other hand, using a minus signimplies knowing where 0 is. The practical consequence is that thevertical translation part K depends on the (artificial) locationof 0. Here it was chosen to put the red and blue starting points in approximately correct relative position.
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