Solution
to Canon
4. a 2 per Augmentationem contrario Motu
Bach also added an inscription, Notulis crescentibus crescat Fortuna Regis
"As the notes increase may the fortunes of the King do likewise."
How
many voices are there?
Hints:
Look
at the title.
“a2”
means that there are two voices in a canonical relationship.
How
many clefs are there?
There
are three clefs:
There
is a C-clef in the upper section.
In
the lower section, there are a C-clef, and a G-clef.
This
shows that there are three voices.
Is
there a
%
?
There
is one in the lower voice, after ½ measures, indicating, together with the
two clefs, that the lower section has two voices in canon with each other.
Which
voice(s) play the Royal theme, and which play the counterpoint?
The
upper voice plays a variation of the Royal theme, and the two lower voices
play a counterpoint in canon.
What
are the transformation principles?
Hints:
Look
at the title and the added text.
What
does “Contrario Motu” mean, (the same as in the last puzzle)?
The
two voices go in opposite directions.
What
are the orientation of the clefs and flats in the lower section?
The
C-clef is normal, but the G-clef is upside-down, which, as in the last puzzle,
also indicates that the follower goes in the opposite direction. This is also
a mirror canon.
Here,
the leader starts by going upwards, and then downwards. Therefore, the
follower starts by going downwards, and then upwards.
On
which note do the leader and follower start, given the two different clefs?
Notice
that the C-clef and the G-clef are centered on the same line, therefore the
leader starts on middle “c,” and the follower starts on “g” above
middle “c.” There is a displacement in musical space.
What
does “augmentation” mean, and what could this indicate as a transformation
principle?
“Augmentation”
means to increase. What could increase?
Think
about the added text – “As the note grow, so may the King’s fortune
increase.” How could the “notes grow”?
The
notes can grow in time. In this puzzle, the notes of the follower are twice as
long as the leader – it moves with half the speed. The result is that the
leader is played twice, during the same time that the follower is played once.
Here,
we have a new transformation principle – a doubling of time.
Is
there a
%
?
There
is a %
at the beginning of the second half of measure one,
indicating that the follower starts when the leader arrives there. Here, we
again have a displacement of time.
Here
we have four transformation principles!:
1.
opposite
directionality
2.
a
displacement of musical space
3.
a
doubling of time
4.
a
displacement of time
In summary, the leader starts on “c,” while the follower starts on “g,” ½ measure later, plays in the opposite direction, and with notes twice as long as the leader!
Geometrical
projection:
Starting
from two superimposed like triangles, or other forms, perform the four
transformations.
How
could you show the augmentation of the triangles?
The
result would look like this:
Performance:
Play
the transformed Royal theme.
Listen
here to the flute: make
MP3
Play
the leader until the beginning of measure 6.
Play
the follower until measure 5.
Play
them both together, if you can, or skip this one.
Play
all three voices, by adding the leader to this recording of the flute playing
the variation of the Royal theme, and the piano playing the follower, up to
measure 6.
