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g(t) = -f(t-0.5) + K and Canon 3
Canon 3 is described by Bach as a 2 per Motum contrarium.
Bach scored this canon in four measures.
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;
Voice 2 starts on G (a perfect fourth lower) at the sign (one-half measure
later) 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 this
example the blue function g is defined from the red function f
by
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.
Back to the ``Math and the Musical Offering'' Main Page.
On to the Next Example.
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