The Apprentice of Fugue, on piano

February 12, 2024

Some of you might already know that I released a new album on February 4, 2024 — a performance of The Apprentice of Fugue played on piano (a Steinway D concert grand). The music was written in 2022, and that year I had released a recording on organ under the original title of the work in German, Der Lehrling der Fuga.

To celebrate this release, my friend Stephen Malinowski made animation videos for Contrapunctus 11 and Contrapunctus 3. As many of you know, Stephen has an uncanny ability to match music to visual patterns. He also knows Bach's music very well, and has an excellent understanding of counterpoint and composition, so he had no trouble at all getting into the inner workings of these pieces. We corresponded over a period of several weeks, during which time Stephen tweaked the visuals again and again. I hope you'll take some time to appreciate the great attention to detail taken in these videos. In Contrapunctus 3, the chromatic lines are rendered with something that looks a bit like the eddies of a small fish swimming in clear water, which fits the feeling of the counterpoint perfectly.

In Contrapunctus 11, there are three subjects, which are the same three subjects used in (my) Contrapunctus 8, but inverted (turned upside-down). Whereas Contrapunctus 8 has three voices, Contrapunctus 11 has four; as in Bach's original, the fourth voice often verges on becoming a fourth subject – a stepwise line, usually chromatic. Bach used repeated notes in his third subject to give a feeling of insistence and urgency. One of my goals throughout this project was to avoid simply copying whatever Bach did. Rather, I wanted to to come up with my own way of doing something similar which might have a chance of being as successful (which I consider a healthy view of tonal composition in general). So I use the notes of his main theme, but (importantly!) I change the meter and rhythm. All other subjects (apart from the BACH subject in the closing fugue) must use different notes. For that reason, my third subject in Contrapunctus 11 doesn't use repeated notes — at least not directly. Instead, neighbour-tones are used. Stephen rendered this subject with spiralling thread-like patterns that fit the contour of the subject perfectly. Bach's second subject begins with a long note on the downbeat, and his third (repeated-note) subject begins with a weak-beat accented note that's often held over a barline (we can call this a "syncopation") — an exciting effect which produces expectation. I turned that around, using syncopation at the outset of my second subject, which allowed me to sometimes approach the third subject using syncopation, in such a way that the listener can't be sure which subject is going to follow that first note. That uncertainty lends an edge to the development. I discussed this with Stephen, and in the video you'll see that those syncopated notes are larger than the others, circled, and they rotate as a sort of "handoff" to what follows. It's these kinds of details that make these videos so much fun to watch!

If you like, you can also watch the corresponding "inversion" of the above triple-fugue in a video of Contrapunctus 8 Stephen made of the first recording of Der Lehrling der Fuga (played on organ, released in 2022).

Stephen also previously made a video of Contrapunctus 9 (which features double counterpoint at the 12th) …

… and Contrapunctus 1, which is among the least similar formally to any of Bach's original fugues, because (as you can read in the liner notes of the recording, or in the score) The Apprentice of Fugue also tells a story, which begins with PART 1 as follows:

"In order to make the masterpiece undoubtedly different, the apprentice looks for alternatives – including potentially clumsy combinations which the master had not used."

Needless to say, I would have never been able to write an Apprentice of Fugue were it not for Bach's Art of Fugue. Which touches on the reason I took on this project to begin with: to learn things I simply couldn't have otherwise learned. There's more to say about that, but I'll end here for now. In any case there will certainly be more episodes of Passages That Bother Me, so stay tuned!

Until then, I remain,
— AAH

Re: Passages that bother me BWV 552a (St. Anne, Prelude)

July 22, 2023

This is a short followup to our previous episode in the series "Passages that bother me", where I thoroughly scrutinised a questionable transition found in the great St. Anne Prelude which opens Bach's Clavierübung III, published 1739. We begin by citing the passage in question as Bach wrote it. The transition in question is between bars 175 and 176.

After publishing my investigations, which contained many possible approaches meant to possibly "improve" the above transition, I received a remarkable alternative which hadn't occurred to me, written by the esteemed composer Stéphane Delplace. A brief exchange of observations led to the following "final version".

The core idea of this alternative is to replace the B-Flat triad in first inversion with a D minor triad in root position. It's difficult to convey just how much wisdom is contained within this surprisingly simple idea.

Recall that a B-flat triad is the most distant diatonic relation to be found in C-Minor, and this fact is in large part responsible for giving the transition an awkward effect. When we replace this harmony with D-minor 7th chord, the majority of the awkwardness is removed in one stroke, since a very distant harmonic relation is replaced with a very close one. When the fifth is present, the chord is an altered harmony in the key of C minor, since the 6th degree must be raised from A-flat to A, but Bach's voice-leading suggests that the fifth shouldn't immediately appear; rather, it should "sneak in" almost unnoticeably. This note, A, then acts as a leading tone to B-flat, so that the voice-leading is finally as smooth as it should be. Because the C is now held as a common tone, the similar motion is decreased just enough that it no longer draws any undue attention to itself.

As an added benefit, this sonority on D minor contains not only a 7th, but also a 9th, namely the small 9th E-flat above the root D. Here is where a deep knowledge of Bach's harmonic language helps to understand why using this specific chord as the solution might just qualify as a stroke of genius! Because this specific chord, a minor triad with a minor 7th and minor 9th, turns out to be one of the most important aspects of Bach's harmonic language. I almost want to say something like "the best kept secret" of his harmonic language, because it seems few people have taken notice of Bach's consistent use of this specific harmony throughout his work. Precisely this sonority occurs as a ninth chord in the major mode, on the third degree, iii9. In minor, it appears here as an altered ii9, where the perfect 5th in the chord is the raised 6th of the minor mode, which is exactly what we see here in the above solution.

In fact, Stéphane Delplace has written a treatise on this very subject, this sonority and the special role it plays in Bach's music, which may be one of the most important treatises on harmony in the modern age. To everyone who wants to understand Bach's harmony, please go now and read this treatise. It is available directly from Delplace's website both in the original French, and in an English translation (which I had the honour of helping with).

III V § I C A (MUSICA, a treatise on harmony by Stéphane Delplace)

As a final note, I also came up with my own version using this same approach, to address a few things about the above solution which still bothered me. First, I found that the B-flat appearing already on beat 2 in the alto was too early, and the arrival should instead be saved for the descent of the soprano. Second, the full triad in three sustained parts on beat 3 seemed to stray unnecessarily from the original which sounds only the single note B-flat at that point. Lastly, the chords in the right hand require an awkward "hopping" motion if they are to be played by one hand, which I found counter to Bach's normal fluid style (although it is easy enough to play the passage by taking the second alto with the left hand, which is how I played it for the above recording). My version resolves these issues by introducing the fifth (A) as a quarter note on the second beat and resolving it on beat 3, resolving the second soprano to a unison with the first soprano, and dropping out the alto on beat four. The result is as follows.

Upon comparing these, I have to admit that, despite my objections, Delplace's version sounds smoother. The reason is that it contains no distracting elements, whereas the accented A in my version makes the ii9 stand out. Therefore I consider Delplace's version to be better. Which do you prefer? As always, I'm curious to hear what you think. Whatever you do, be sure to get the treatise! It has the potential to help you appreciate Bach's music on a new level, and radically change the way you think about harmony in your own work.

Peace,
Aaron

Passages that bother me: BWV 552a (St. Anne, Prelude)

May 15, 2023

Today I welcome you to another episode in the series "Passages that bother me". It's very rare that I have the feeling that anything should be changed in a piece by Bach, but loyal readers of this blog will know that I've written about an example or two in the past. This time it's the Präludium pro Organo pleno, from BWV 552, the great St. Anne Prelude and Fugue which is found at the opening of Bach's Clavierübung III, published 1739, which contains a passage that bothers me.

The Structure of the Prelude

In some ways the form of the prelude resembles the scheme of a French Overture — a ternary form (we'll call it ABA) in which the opening and closing material (the "A" material) is slow and stately, with dotted rhythms and scale runs, and the middle section (the "B" material) features fugal imitation. In this case, instead of ABA, we have a five-part structure ABABA. This 5-part concept is a gross oversimplification of the form (completely ignoring secondary thematic material), but it serves the purpose at hand.

The piece opens with a clear question / answer phrase structure, using a descending scale motive embedded within chords in half notes which cadence on suspended harmonies. [N.B. The embedded audio clips in this blog post are live recordings of me playing the St. Bavokerk Müller organ sample set (Voxus) using Hauptwerk.]

We'll call this the "A" material. After some sequencing, a similar phrase structure takes place in bar 17, beginning on the dominant (B-Flat), this time moving to the relative minor (C-Minor), also with suspended harmonies.

NOTE: The passage above becomes important for understanding how Bach transitions from the "B" material back into "A".

There is also a subsequent varied statement of this question / answer harmonic pair developed using more active scale runs, helping to warm us up for the B material, which is more active, consisting of shorter note values (16ths) accompanied mostly in quarter notes. The B material is developed in such a way that we become "swept away" as it were, so that we might even almost forget where we came from, and the return of the opening material becomes all the more pleasant.

The Art of the Transition

Anyone who attempts to compose a piece of music in which multiple themes are supposed to appear will find out how challenging it can be to transition from one theme to another. In such a piece, assuming the material is solid, the trickiest parts to write are likely to be the transitions. In the baroque style, transitions are usually done using cadences. In this prelude, there are four significant transitions which take place between the A and B material. The first, moving from A to B, is found at bar 71, a cadence in C-Minor.

The subsequent transition back into the A material is made to sound natural by immediately engaging in a sequence after a suspended harmony on F-Minor in bar 99. Recalling our note above, you may notice that this A material is a transposition of the sequence which started in bar 20.

So this turns out to be a transposition of the opening material in the subdominant, where the transition enters at the equivalent of the relative minor suspended harmony we saw in bar 20 (before: C-Minor in the key of E-Flat, here: F-Minor in the key of A-Flat).

Later, another transition into an extended treatment of B appears with a V I in E-flat (the home key) in bars 129-130.

The B material is then developed over a span of 103 measures without return to the A material, until bar 175, where the half notes reappear, not in E-Flat major, but in C-Minor (the relative minor), followed by a complete restatement of the opening (around 30 measures).

Measures 175-176

And here we have arrived at the passage that bothers me, which has in fact bothered me for as long as I can remember. Every time I hear this piece or play it myself, the shift from bar 175 to bar 176 sounds too abrupt — almost like a "cut and paste". It seems like the A material was simply put there without enough care being taken for the transition. Every time I experience this piece (and I keep hoping the feeling will go away, but it never does), I feel like something has gone a bit wrong and something is missing at this moment. Why? Is this just some personal quirk of mine? I don't think so, because there are logical reasons why it sounds not quite right. I will even go out on a limb here and wager that in the 300 years since it's publication, at least one other person has been bothered by this transition.

If we compare this passage from bar 174 to the end of the piece with the previous transition from B to A in bar 99 and following, we notice that this second transition would be symmetrical with the first if Bach made the "splice" from bar 174 to bar 193 instead of bar 175, because in both bars (175 and 193) we have essentially the same suspended C-Minor cadence half notes. Did you ever notice that before?

If this transition were to move from bar 174 to bar 193, the closing A section would be exactly symmetrical with the transition in bars 99 and following, and the final section would be 18 bars shorter.

Could this shorter recapitulation have been what Bach started with, and then he decided it was better to restate the whole opening? Not likely. A lesser composer might have done that, but not Bach. Bach knew that the transition would have been much more natural if it had moved from 174 to 193, because that's what he did at bar 99, but if he did that again here, the formal balance wouldn't be right. By including 18 additional bars from the opening, the form is correctly balanced, but the transition is slightly awkward. Bach knew that front-to-back formal symmetry was more important than the transition. In fact, the "surprising" aspect of the transition could in this light be viewed as a deliberate signal to make the formal return of the opening material all the more clear.

While that sounds reasonable enough, the transition still bothers me. Here's why. Before the return of the A material in bar 175, C-Minor has already been established 6 measures earlier (with the B material), so the note B-natural as the leading tone has already been in clear force prior to the transition.

When we reach bar 175, we get a very strong suspended harmony, V over i resolving to i in C-Minor, and the crucial note B-natural sounds in the V resolving to C. In the following measure, the harmony shifts to B-flat Major in first inversion, with almost all voices moving in similar motion. If those two facts seem like clues that something is less than ideal here, you're starting to see why this bothers me.

A Questionable Harmonic Shift

B-Flat Major is in fact a functional diatonic harmony in C-Minor, but it's also the most distant relation, the flat VII. B-Flat Major sounding after a very clear C-Minor in which half of the bar has sounded the leading tone B-natural in the dominant V and resolved it, simply isn't going be the most natural-sounding transition (even when the preceding referenced the key of A-Flat, with the note B-Flat appearing in bars 171-173). The harmonic shift from C-Minor to B-Flat is by nature going to sound a little less than natural, if not slightly awkward, because of the chromatic shift from B-Natural to B-Flat.

Questionable Voice-leading

Too much similar motion is not good for counterpoint, and exactly that weakens this crucial moment. The D in the bass is supposed to be a leading tone in E-flat Major, but that isn't what it sounds like when it arrives. Instead, we are distracted by the chromatic shift from B-natural to B-flat (downward!) which works against the D sounding like a leading tone needing to resolve up by step. The soprano, alto, and tenor all move up. The bass technically steps down, but its line in bar 175 is a neighbour group around D, so the impression is still vaguely of stepping up from the downbeats of these two measures: C to D.

Alternative 1: A First Attempt

Could this transition be improved? Is there something "missing" here? To try to answer these questions, I wrote some alternative transitions. The first attempt was simple, using the advice "don't worry about anything, just follow your ear" (which is good for getting a quick result — not necessarily the best result, but a quick one). So I began with the piece up to bar 175 as written, and imagined that the score broke off there, with some music missing, then picked up again in bar 176. What might be missing? Again, the goal of this first attempt was not to write the greatest not-actually-missing bars in history. It was to have something that did the job, which could then help give clues about how to perhaps do it better. Following my ear wherever it led, I ended up with the following five measures:

I understand that at this point you might feel like telling me how awful this is, but let's look at what happens here objectively if we can, and see how it relates to the original. Instead of directly cancelling the B-natural leading tone with a B-Flat, A-Flat gets raised to A-natural so that the B-Flat sounds more natural along with a raised F-Sharp, to lead to G-Minor. This bass line is reasonable voice-leading, since it is in contrary motion with the soprano (and is the same bass pattern Bach used at the opening of the work — a downward tritone). The same harmonic progression then happens in D-Minor (a sequence), but with a thinner texture and a deceptive resolution of the dominant. Then comes a quick step backward on the circle of fifths, and a statement of the B theme appears in B-Flat (with a "hand-off" between the soprano and tenor), with another unexpected backward fifth modulation so that B-Flat clearly becomes the dominant of E-Flat.

This works, but feels a bit long, and has a harmonic direction that may not be entirely convincing. The cadence in G-Minor seems suspect. The problem is that that the symmetry of two suspended half-note harmonies of the opening theme is broken, since we now we have three such suspended harmonies (C-Minor, G-Minor, E-Flat). This raises a point in favour of Bach's version: simple symmetry. There are other issues within these five bars. Below, I've marked my score as if I were one of my students.

If anyone out there is concerned about the implied parallel 5ths between the tenor and bass in bar 180 above, take a look at my previous blog post which shows how Bach wrote implied parallel 5ths and implied parallel octaves, as well as one actual parallel octave / unison in this piece.

Alternative 2: A Second Attempt

Considering all of the issues in my first try, I made a plan for a second attempt, aiming for something as short as possible, trying to keep whatever might have been good in the first attempt. The result was the following two measures.

Here, the bass line from bar 175 to 176 remains as Bach wrote it, and the soprano line in bar 176 is almost the same except that what was a B-Flat is here a B-natural, and the harmony is D-Half-Diminished, so we stay in C-Minor. A deceptive resolution of V then moves to A-Flat, followed by a ii V I in E-Flat. Harmonically, this is more logical than the first attempt, and it seems to do the job without raising any obvious red flags. The symmetry of the original is still sort-of there, delayed by one measure.

This version sounds less invasive than my first attempt, but the fifths by contrary motion in outer parts bothers my ear, and I'm also left with the feeling that there isn't enough space, because we plow through with all parts sounding instead of dropping out voices as in the original. So another important aspect of the original becomes clear: space!

Another apparent weakness of this version is that we've just cadenced with a 2-5-1 in C-Minor, so the 2-5 (D-Half-Diminished to G) in the new bar 176 ends up sounding slightly repetitive.

Alternative 3: A Revised First Attempt

Returning to the first attempt to try to improve it, I noticed that the cadence in G-Minor could be removed, the deceptive cadence could be sequenced instead, and the soprano line transitioning into the suspended harmony in E-Flat could be altered in a way that would both make the melodic direction of the phrase more logical, and also produce a broader symmetry within the composition, so I made those revisions, as shown below.

With the cadence in G-Minor removed, the harmonic rhythm is certainly more consistent. The transitioning soprano line in the last bar now begins on A-Flat, higher than its previous entrance, and is now modelled after the alto in bar 174, creating a new symmetry. On the whole this should give a more natural and convincing impression, but it remains quite far away from the original.

At this point another important aspect of the original became clear to me, namely that the entrance of the pedal (which has been absent for about 15 measures prior to bar 174) is associated with the return to the A material. Here we don't get that same signal. Foiled again?

Alternative 4: A Revised Third Attempt

Since I saw that none of my alternative ideas matched the straightforward simplicity of the original, I changed my focus to the four measures 174-177 which serve as an analog to the opening four bars of the work. Those opening four bars are shown below, followed by bars 174-177.

It occurred to me that without adding any new measures, a simple change in harmony could bridge the keys of C-Minor and E-Flat Major in a more functional way, while also improving the voice-leading. The trick would be, instead of moving directly to B-Flat, the bass would move to A as the 3rd of F Major, which would then move to B-Flat in first inversion on beat 3. By holding only the B-Flat in the soprano as a dotted quarter and making all the other parts on beat 3 quarter notes followed by eighth rests on beat 4, the feeling of rest before the suspended harmony on E-Flat could still remain half-preserved. Here is the result of this thinking:

F Major is functional in both keys, serving as a more reasonable pivot than B-Flat. This option somehow manages to sound less convincing than the other attempts. One reason for this may be that Bach tended not to write two first-inversion 5th-related triads in a row where the bass sounds like it should be in root position but it isn't. Another reason mentioned earlier would be the fact that all voices remain present. It also contains implied octaves by contrary motion. Otherwise it's possible that the this version may be too similar to the original (which for those of us who know this piece is burned into our brains), so that cognitive dissonance is too strong, and it just "sounds wrong". Which begs the question, for someone who didn't know this piece to begin with, how would this version fair against the original?

Alternative 5: A Completely Different Approach

After all this bother over measures 175-176, where C-Minor is the given which causes so much trouble moving to B-Flat, it dawned on me that the sequence prior to these measures could simply be redirected so that we don't move to C-Minor in the first place. Instead, the sequence can move to B-Flat Minor, using a Phrygian half cadence. The trick is to use D-Flat, G-Flat and B-Flat instead D, G and B, since the sequence itself can naturally move in either direction. Once we have an implied F dominant instead of a G dominant, we can cadence in B-Flat Minor instead of C-Minor. Then the B-Flat dominant chord functions more strongly in both keys, as an altered tonic (secondary V⁷ of iv) in B-Flat Minor, and as V⁷ in E-Flat. An added 7th in the tenor on the dominant strengthens the shared harmonic function. Spoiler alert: this is going to sound a bit clumsy!

The clumsiness comes from two things: first and foremost, the single voice trailing off (as the original does) doesn't have quite enough context to state a totally convincing Phrygian half cadence. Secondly, the soprano stepping down from C to B-Flat instead of B is slightly conspicuous. The good news is that these things can be improved. Adjusting things slightly to move to B-Flat sooner, leaving in the other parts at the cadence, gives a better result. There would be any number of ways to work out the voice-leading for this; below is one possibility.

Although the harmony is clear and functional, the feeling I get from this version is a bit painful. B-Flat Minor is a lovely but rather torturous key in relation to E-Flat Major, the resolution on the high D-Flat sounds slightly too strident, and the chromatic shift to B-Flat dominant with the 7th in the tenor is too distressful at this moment.

So another trait of the original appears to be superior in comparison. Above, the soprano line steps down from E-Flat to G, but has to do an intervening chromatic shift from D-Flat up to D. Bach's version simply steps down diatonically from C to G — much simpler.

It would be possible to resolve the suspended harmony to B-Flat Major instead of B-Flat Minor to avoid the upward chromatic shift during the descent, but where do we stop? At this point, I think the lessons have been learned.

Conclusion

Are any of these alternatives an "improvement" on the original? No, I wouldn't go that far, although when I play the prelude using any of the more successful attempts, the awkwardness of the original harmonic motion and voice-leading is gone, and for me, that's a better feeling than I get from the original. That might not be the case for you, or for anyone else on the planet, but that isn't the point of the exercise. I've learned quite a few things about why the original is the way that it is, by figuring out some of the things that make it good in comparison to possible alternatives. Obviously, if I hadn't made the effort, I wouldn't have learned these things. That's justification enough for allowing myself to "rewrite" the music of the master. There are those who will protest, claiming that what I've done here is disrespectful. I've received these kinds of comments on social media many times before: "Who are you to ruin this music?!", "Disgraceful!", etc. Let's please not be so uptight. Rewriting Bach's music is a very good way to learn.

Was everything I learned from this exercise already obvious to you? If so, then congratulations for being a few steps ahead of me! If on the other hand you learned something from what I've done here, I'd be glad to hear that. Especially if my experiments inspire you to come up with your own alternative versions of passages that bother you, I hope you'll share those. And lastly, by the way, if you can't think of any passages that bother you, you may not be listening closely enough …

Peace,
Aaron