Tetrachords and Bach: Not quite a primer (pt.1)

Stumbling through a piano reduction of Contrapunctus VIII from Bach’s Art of the Fugue, I found myself returning over and over again to this short passage that occurs in the bass voice in measures 111-113:

1. From Contrapunctus VIII, sequence of ascending 3rds and descending 2nds.

1. From J. S. Bach’s The Art of the Fugue, Contrapunctus VIII. A sequence of ascending 3rds and descending 2nds.

This phrase doesn’t have any relevance to the fugue’s three subjects, so referring to it as a motive would not be accurate. Individually, these four-note patterns occur repeatedly throughout, but this is the only instance where they occur in a consecutive sequence of this length and in a single voice. This type of sequence, consisting of ascending 3rds and descending 2nds, is a common sight in much of Bach’s music, piano or otherwise. It contains a significant amount of both harmonic and melodic information, and because of that, this is as good a starting place as any to begin making sense of tertachords. Below, I’ve re-ordered the two inner notes of each group so that they now ascend diatonically in a step-wise manner.

2. Notes re-arranged to appear in a simpler form as tetrachords.

We could call them tetrachords even in their original ordering, but it’s visually much easier to recognize them if they appear in a step-wise sequence. It would be an impossible task and likely diminish the importance of tetrachords by attempting to define or explain them in a single post, so let’s start with a definition that is simple, admittedly inadequate, and with the intention to elaborate upon it later: A tetrachord is a group of four consecutive notes that make up either the lower or upper segment of a scale or mode and whose sum is equal to the interval of a fourth.

Continuing to take some liberties with this passage, the image below has been re-arranged again in octave transpositions, making the interval of the 7th that previously separated each group from the next seem like a much less dramatic distance. Now, not only do the four-note groupings appear in step-wise spellings, we can see that the entire phrase might simply be a scale whose combined ascending range is only slightly greater than two octaves.

3. After octave transpositions, the passage appears as a diatonic scale.

To see this with an even clearer illustration, here is the passage again with the key signature and the accidentals removed:

4. Passage without key signature or accidentals.

For the sake of making what I hope is a reasonable introduction to tetrachords, I readily admit to the purposeful omission of some very important considerations regarding the process outlined above. For instance, it doesn’t really make sense to isolate this passage from Bach and attempt to simplify it. The notes have been altered to suit the needs of an argument that is likely unrelated to the intended context of the passage. The harmonic considerations have been completely ignored, at least for the time being, and there is no mention of any performance indications, the rules of counterpoint, or to the structure and design of this particular fugue.

Anyway, I have to start somewhere.

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