The Art of the Fugue
Canon at the Twelve (BWV 1080/17)
The Canon alla Decima in contrapunto alla Terza is an excellent example of the use of the inversion at the 10th technique. If you need help understanding the contrapuntal inversion technique read Invertible Counterpoint. Let's take a look at the formal structure of this canon before looking into the invertible counterpoint issue:
The canon begins with an 8 measures theme or subject. Notes in red are those from the original theme in which this canon and all fugues are based:
The second voice begins in measure 9 but the imitation is at the 12th or 5th:
Listen to the first 16 measures:
In measures 17 to 33 the upper voice continues the imitation of the lower voice. In measure 34 it presents the subject:
From measures 34 to 41 the lower voice will add a new counterpoint. From measure 42, measures 9 to 33 are repeated but with the voices inverted at the 12th. The lower voice transposes two 8ves higher and the upper voice one 12th lower. Because of the type of inversion used, the lower voice will now imitate the upper voice at the 8ve and not at the 12th as in the first part.
The upper voice in measure 7 becomes lower voice in measure 42 but transposed one 12th lower (A becomes D).
The lower voice of measure 7 is transposed two octaves higher in measure 42.
In measures 68 a 75 the lower voice presents the subject. The upper voice uses the counterpoint from measures 34 to 42, but transposed a 12th higher. A short coda (76 - 78) ends this canon.
A short summary of the structure:
|1 - 33||The upper voice imitates the lower voice one 12th higher|
|34 - 41||The subject reappears in the upper voice while the lower voice presents a new counterpoint to the subject|
|42 - 67||The lower voice begins imitating the upper voice using the inversion at the 12th.|
|68 - 75||The subject reappears in the lower voice. The counterpoint from measures 34 - 41 reappears in the upper voice using inversion at the 12th|
|76 - 78||Coda|
Listen to the complete canon. We have highlighted the imitations using numbers. For example, the subject is labeled as 1 and when imitated by the upper voice, the upper voice imitation will be labeled as 1. We also show the intervals between voices:
Invertible counterpoint in the canon at the 12th
Inversion at the 12th table
We will now analyze how Bach works out the problems related to the use invertible counterpoint. In invertible counterpoint at the 12th the 6ths intervals are problematic because after inversion they become 7ths. How does Bach handles this problem?
After an analysis of the intervals used by Bach, we find that he uses mostly 3rds, some 5ths and 8ves and very few 6ths. Clearly he tries to stay away from the problematic 6th interval as you can see in the previous example.
Some examples using sixth and sevenths:
The way Bach works around the problems with the 6ths and 7ths intervals is using the 7th interval as a suspension or as part of a dominant chord harmony. Once inverted the resulting 6th interval presents no problem.
Measures 11 and 44
The 7th interval in measure 11 is used as a suspension. No problem is caused when it inverts in a 6th in measure 44.
Measures 13 and 46
The 7th in measure 13 is the 9th of the E dominant chord. When inverted the note becomes the 7th of the C# diminished seventh chord.
Measures 16 and 49
Once again the 7th in measure 16 is used as a suspension.
These are good examples of how you can use 6ths and 7ths in invertible counterpoint at the 12th.