Art of the Fugue
Canon alla Decima in contrapunto alla Terza (BWV 1080/16)
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.
This canon begins with a 4 measure subject or theme:
This subject is based on the original subject transformed by contrary motion or inversion (ascending intervals become descending intervals and vice versa). In the following image we compare both subjects:
In measure 5 the second voice answers the bass at the 10th:
The upper voice follows the lower voice until measure 40, where it presents the subject:
Yet there are some interesting details before measure 40. Measures 5 to 8 are inverted at the 10th in measures 9 to 11:
As in the canon at the 12th, Bach repeats measures 5 to 39 but inverting the voices at the 10th.
The canon ends with a small coda in measures 79 to 82. A short summary of the structure:
|1 - 39||Lower voice is imitated at the 10th|
|40 - 78||The subject reappears in the upper voice while the lower voice begins imitating the upper voice using the inversion at the 10th.|
|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 10th
Inversion at the 10th table
As you can see in the inversion table at your right, the inversion at the 10th presents a lot of problems. The most useful intervals - 3rds and 6ths - invert as 8ves and 5ths. This means that if you use parallel 3rds or 6ths you will get parallel 8ves and 5ths after inversion. Also, if you use a lot of 3rds or 6ths you will get a lot of 8ves and 5ths after the inversion and your counterpoint will probably sound too consonant and uninteresting.
When we analyze the counterpoint in this canon we find that Bach:
- Uses contrary motion whenever 3rds or 6ths are used to avoid parallel 8ves and 5ths.
- Makes a balanced use of imperfect consonances (3rds or 6ths) and perfect consonances (8ves and 5ths).