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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 intervalsbecome 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 9to 11:

As in the canon at the 12th, Bach repeats measures 5 to 39 but inverting the voices at the 10th.

Inversion at the 10th table

2 3 4 5 6 7 8 9

9 8 7 6 5 4 3 2

The canon ends with a small coda in measures 79 to 82. A short summary of the structure:

Measures

1 - 39 Lower voice is imitated at the 10th

40 - 78The subject reappears in the upper voice while the lower voice begins imitating theupper 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 islabeled as 1 and when imitated by the upper voice, the upper voice imitation will be labeled as 1. We also showthe intervals between voices:

Invertible counterpoint in the canon at the 10th

As you can see in the inversion table at your right, the inversion at the 10th presents alot of problems. The most useful intervals - 3rds and 6ths - invert as 8ves and 5ths. Thismeans that if you use parallel 3rds or 6ths you will get parallel 8ves and 5ths afterinversion. Also, if you use a lot of 3rds or 6ths you will get a lot of 8ves and 5ths afterthe inversion and your counterpoint will probably sound too consonant and uninteresting.

When we analyze the counterpoint in this canon we find that Bach:

1. Uses contrary motion whenever 3rds or 6ths are used to avoid parallel 8ves and 5ths.2. Makes a balanced use of imperfect consonances (3rds or 6ths) and perfect consonances (8ves and 5ths).

Recordings:

Robert Hill, harpsichordKeller QuartetHesperion XX - Jordi Savall

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2004 Jos Rodrguez Alvira. Published by teoria.com