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:

Canon 4 animation


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.

Recording here.

To canon 5