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 voices begins in measure 9 but the imitation is at the 12^th or 5^th:

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 12^th. The lower voice transposes two 8^ves higher and the upper voice one 12^th lower.  Because of the type of inversion used, the lower voice will now imitate the upper voice at the 8^ve and not at the 12^th 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 12^th higher. A short coda (76 - 78) ends this canon.

A short summary of the structure:

Measures
1 - 33	The upper voice imitates the lower voice one 12th higher
34 - 67	The subject reappears in the upper voice while 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 - 42 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:

Invertible counterpoint in the canon at the 12th

We will now analyze how Bach works out the problems related to the use invertible counterpoint. We know that 6^ths intervals are problematic because after inversion they become 7^ths. How does Bach handles this problem?

After an analysis of the intervals used by Bach, we find that he uses mostly 3^rds, some 5^ths and 8^ves and very few 6ths. Clearly he tries to stay away from the problematic 6^th interval.

In the following example you can take a look at the complete canon with the intervals indicated between staves:

Some examples using sixth and sevenths:

The way Bach works around the problems with the 6^ths and 7^ths intervals is using the 7^th 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 7^th interval in measure 11 is used as a suspension. No problem is caused when it inverts in a 6^th in measure 44.

Measures 13 and 46

The 7^th in measure 13 is the 9^th of the E dominant chord. When inverted the note becomes the 7^th of the C# diminished seventh chord.

Measures 16 and 49

Once again the 7^th in measure 16 is used as a suspension.

These are good examples of how you can use 6^ths and 7^ths in invertible counterpoint at the 12^th.

Related links:

• Complete score in IMSLP / Petrucci Library
• Recording
• The Art of the fugue in Wikipedia