Is this bad invertible counterpoint? (BWV851)

January 19, 2023

One of my students recently asked an interesting question about invertible counterpoint. He pointed out a passage in Bach's fugue in D Minor from the Well Tempered Clavier, book 1 (BWV851). In measure 3, there are 4ths between the soprano and alto (the only two sounding voices) on beat 4 and the 'and' of beat 4 (the 4ths are marked in red).

The same configuration happens again in bar 6 between the soprano and bass, but it's not necessary to show that, since this configuration was not cause for the question. The question comes from the observation that Bach uses these same patterns in inverted positions in bar 13. In other words, this is an instance of typical double counterpoint at the octave. Do you notice that things look a little suspect here?

Here in bar 13, we see what looks like implied parallel 5ths between the soprano and bass! What's going on? Isn't this bad invertible counterpoint? How did Bach get away with this?

If we take a closer look at bar 3, we notice that both of the 4ths are accented dissonances. The harmonic context makes this clear. The F and the D, both consonant 3rds, are the tones which determine the harmony, not the G and the E (the dissonant 4ths). Here the red notes are showing consonances and the dissonant notes / intervals are marked in grey.

To stress this point, we can imagine an opening where in bar 3 the accented dissonances have simply been removed.

Of course, Bach didn't write that. But that is what's underneath what Bach wrote. When Bach inverted the parts as written in bar 13, it's a similar situation. The offbeat 6ths determine the harmony, not the 5ths.

This is, we must admit, an extremely precarious thing to have done! The middle voice A has been circled, because without that note, the whole structure falls apart! In two parts, this inverted form would simply not fly.

The 5ths are way too conspicuous when we have only two parts. A clear no-go. The structure needs that A in the middle voice, not only to clarify the harmony, but also to draw attention away from the implied 5ths by sounding in close dissonance with the bass (a clashing 2nd). A very clever move by Bach! (If you like clever things, you'll love Bach).

By the way, the markings shown above are what I would normally use when giving feedback to students. Notes, voice leading, patterns, etc. which need attention all get marked, and each point gets a red letter in parentheses, which then is given a thorough explanation (usually the alphabet supplies enough letters for the feedback, but I have in fact had cases where double letters were necessary).

You may be wondering though, how do we explain exactly what is going on harmonically here? Well, it may seem slightly odd, but it's actually pretty typical of Bach. We can add a couple of notes to arrive at a more conventional version which clarifies the harmony.

[N.B. The chord symbol Dm should be Dm/F] Bach didn't write that G-sharp, so the real harmony there is a "suspended" sound, but functionally that middle voice begs to sound a G-Sharp at that point, giving a second inversion diminished triad. If the voice were to continue sounding, then it would return to A on the downbeat as shown, resulting in a second inversion A Major triad. Do these clearer harmonies raise any red flags? Yes. Second-inversion flags. This is a pretty logical explanation as to why Bach didn't write these notes. Although the 64 on the diminished chord is passable, the downbeat 64 is very undesirable. We could say it's still there in Bach's actual writing, but the A isn't there, and a 6th without a 4th suggests an alternate root on the 3rd: C-Sharp to E. This is a trick of the trade you can also use to get around some cases of second inversion harmony. Use a rest, omit the 4th, and the sounding 6th is often acceptable, as it is here.

Bach was very careful in this piece not to repeat this configuration of voices. Bar 13 is in fact the only place in all of 44 bars this situation appears. The voices appear only one time again in a similar structure, in measure 27, in only two parts, but there Bach uses the inversion of the subject so that the structural tones and added dissonances form an entirely new constellation.

Much more time could be devoted to this. It would be easy to spend hours getting into the details of this piece. And it would be time well spent, very rewarding study. That's how Bach's music is. In the whole WTC, from one piece to the next he does such daring things, and if you look you'll find instances that just shouldn't work, but they do. Including parallel fifths! They are there in the WTC. I won't tell you where. See if you can find them!

Until next time,
AAH