On Fugues

20 April 2025

We’re going to start another series of musical notes, this time focusing on counterpoint, and more specifically fugues. We are going to focus on the structure of the fugue itself and counterpoint writing, so I am going to attempt to do a full analysis for each of the pieces.

Bach, Fugue in C minor from WTC I, BWV 847

What better way to start with discussing fugues than the preludes and fugues of the Well-Tempered Clavier. This is a simple fugue in three voices, which is the textbook standard for the simplest fugue possible.

Definitions

• Exposition: The beginning section of the fugue, where every voice gets a statement of the thematic material in alternating subject/answer format.
• Episode: The intervening material between full entries of the fugal material, often composed of fragments of thematic material itself.
• Stretto: The technique where the thematic material (usually the subject) enters in close proximity (in terms of time), achieving the effect that the subject collapses on itself.
• Mid-entry: A restatement of the thematic material in other keys, often used as a developmental tool to explore different harmonies.
• Recapitulation: A final restatement of the thematic material, in the original key, and concludes the fugue.

Bach, Contrapunctus I from the Art of Fugue, BWV 1080

This is the standard example of a simple fugue in four voices, so every entry of the subject can now potentially enter four times, instead of three like the last fugue. There isn’t much to analyze because the structure is relatively simple: in the video, the boxes and the running cylinders highlight the subject/answer, while the cylinders without the boxes represent the fragmented versions of the theme.

There are many more four-part fugues that are worth studying over this particular one, since this has little to no harmonic variation; however, I included this one mostly because the Art of Fugue motif is going to be featured multiple times in this article, so we should be familiar with it.

Ravel, Fugue from Le Tombeau de Couperin

A three voice fugue that has distinctly the mark of Ravel on it. I could talk a lot more about the Tombeau as a whole, but allow me to focus on the fugue itself today. Most importantly, the secondary motif also acts as a countersubject. Notably, Ravel plays with these two elements in a variety of ways, for example, inverting them from the descending motion in the beginning, to the ascending motion as stated in the middle of the piece.

Definitions

• Inversion: The act of mirroring a theme/motif across some pitch of symmetry. Note that there is a more specific notion of inversion in counterpoint, which we shall discuss later.

Beethoven, Finale from 29th Piano Sonata, “Hammerklavier”

This monster of a sonata is one of Beethoven’s greatest works from his late period, alongside his 9th symphony and the late quartets, which mostly embrace the Baroque style of music writing, and paired with Beethoven’s penchant for rhythmically-driven motifs. Of the baroque forms, the most complicated one of course is the fugue, which we have an example of here.

The three-voice simple fugue starts at 2:48, and the subject is comprised of two main elements:

1. the leaping 10th and the trill, which grabs the attention of the listener by interrupting the almost perpetuum mobile texture throughout the movement, represented here by the connected circles;
2. the descending sixteenth note figure and its subsequent scales, which contributes to the perpetuum mobile texture, represented here by the tumbling squares.

While there are many full re-entries of the subject material, none of them can truly be considered a full mid-entry, because only one voice ever enters at a time. That is, until 4:37, where all three voices present the subject, but in augmentation, and occasionally in inversion too.

The next mid-entry is very much hidden from the casual listener, at 5:57. The subject is presented in retrograde, which without the trill grabbing your attention, it is not immediately obvious that there is counterpoint happening.

Normally, this is when a fugue ends, with a final entry, but Beethoven never ends a piece the normal way; he drags it out, and teases the audience with fragments of a code multiple times before he actually presents it. Hence, at 8:24, Beethoven presents us with a small interlude in the form of a fugato, allowing the listener to take a small break from the unceasing pace of the fugue. After that, he presents a small fugue with three entries, before giving us the real coda, bringing back the peaceful chords that started the prelude.

Definitions

• Augmentation: Lengthening the duration of each note, such that the subject is slowed. diminution: the opposite of augmentation, which shortens the duration of each note, such that the subject is quickened.
• Retrograde: Running the subject in reverse. rarely used technique, as it is very difficult to write a subject that works both ways.
• Fugato: A small, fugue-like passage within a bigger piece. It is often written not up to the standards of a full fugue, but still includes hefty amounts of counterpoint. Beethoven loves his fugato, usually inserting it between larger fugues.

Liszt, Fantasia and Fugue on B-A-C-H

In hindsight, after a few listens, the academic quality of studying this fugue isn’t much, but as with all Liszt, it sounds very impressive.

The Fantasia is comprised of the B-A-C-H motif introduced over and over again, at multiple registers and speed, often simultaneously. The fugue is a very much chromatic one, with a very unclear tonal center.

Liszt originally wrote this for the organ, which sounds musically much more “full”, but the piano version is a better one to study the fugue. I suggest you look that version up for the better listening experience.

Bach, Fugue in D-sharp minor from WTC I, BWV 853

This is (hopefully) last of the simple fugues that we are going to cover. With the exception of retrograde, Bach uses almost every technique that we have discussed in the past few weeks. The entry at 1:00 is stretto, and there is already mild amounts of augmentation and inversion (at 1:08 and 1:24 respectively) immediately after. The first mid-entry is purely inverted, and includes a stretto including all three voices. The second mid-entry is a stretto between the subject and its own augmentation. Try to catch the rest of the techniques used!

Mozart, Kyrie Eleison from Requiem

Largely considered to be Mozart’s last work, he envisioned this to be a grand requiem mass, with fugal endings for each of the sections, such as the Kyrie fugue and the Hosanna fugue in Sanctus. Sadly, he was unable to complete this work before his untimely death at the age of 35.

The Kyrie appears as a double fugue, with both themes introduced simultaneously. The Kyrie eleison theme, highlighted yellow, uses the cross motif (trace out the first four notes); while the Christe eleison theme, highlighted green, provides the driving motion for this entire movement. The rest of the fugue is easy to follow, with alternating entries of the two themes until the second mid-entry, where the Christe eleison theme starts appearing in stretto.

Beethoven, Grand Fugue

Originally composed as the finale to his 13th quartet, it had been described by Beethoven’s contemporaries as “an indecipherable, uncorrected horror” (Spohr, 1826). As a result, Beethoven composed a more digestible finale to the quartet (not before he called his audience “cattle! asses!”), and decided to publish the fugue as a separate work.

While the piece is called the great or grand fugue, it is actually a double fugue, with a fugato nestled between the two fugues; and as a result, it has three big themes, one for each of the fugues and the fugato.

The piece starts with an overture, which introduces us to all the themes in reverse order: firstly, a simple statement of theme 2, then demonstrating its use in fugue 2, and then demonstrating how it is used in the fugato (including a small snippet of the fugato, highlighted in yellow), before stating it as the countersubject of fugue 1.

We then get to the first fugue, by introducing theme 1, highlighted in red. It is comprised of jumps of large intervals, and is presented in iambic rhythm; theme 2 is presented in syncopation as a countersubject. In each subsequent entry, Beethoven introduces a new rhythmic device for us to play with. The 1st mid-entry arms us with triplets, which when played against the iambs, is mildly 3 against 4. The 2nd mid-entry is filled with dactylic rhythms, and theme 2 is now played off a main beat, leading to what I call “rhythmic armageddon”. The final entry is presented in diminution, with each of the themes stated in different rhythms again, before breaking down into an uncertain G-flat major chord that each voice slowly arrives to.

The fugato theme is then presented once plainly, then contrapunctally in three entries, in increasing harmonic depth. The theme is then played in unison, before fading out into nothing.

We are then surprised by the appearance of an interlude, preceding the actual second fugue. Beethoven then pits theme 2 against itself in 3 different ways, presenting the subject in constant stretto. As with the first fugue, the 1st mid-entry arms us with triplets, and theme 1 comes back into the counterpoint in the 2nd mid-entry. This all accumulates into the final entry, which is played in constant fortissimo, before leading us back into the fugato theme.

Beethoven usually likes to extend his codas, teasing his audience with the ending before it actually really comes along. In true Beethovenian fashion, we get a section of uncertainty, before jumping right back into the interlude; then another section of uncertainty, before jumping into a snippet of fugue 1 and a snippet of the fugato, and the overture then returns with theme 2 played in 4 octaves. After one more section of uncertainty, theme 1 is then placed lightly atop a simple statement of theme 2, before the grand fugue ends in a glorious B-flat major chord.

Hamelin, 12th Etude “Prelude and Fugue” from 12 Etudes in All the Minor Keys

A harmonically complex double fugue, he plays with two main fugal themes (introduced at 2:44 and 3:36). There isn’t much to remark, other than the fact that there is so much technique involved in this modern fugue.

Bach, Contrapunctus VIII from the Art of Fugue, BWV 1080

First of the triple fugues here, the three themes are developed within the same exposition, resulting in a relatively short fugue. The first theme is given by the hollow rectangles, which is particularly chromatic among baroque themes; the second theme is introduced at 0:56 during the first mid-entry, highlighted by the rhombi, and accentuates the chromaticity of the first theme. And of course, no fugue in the Art of Fugue is complete without its famous theme, of which a variant is introduced at 2:03 as our third theme, highlighted by the cylinders.

Bach, Contrapunctus XI from the Art of Fugue, BWV 1080

I have always meant for this fugue to immediately follow VIII, mostly because all three themes from VIII have been inverted here. Amazingly, this creates a vastly different fugue with a different texture. The three themes are highlighted with the same shapes, so I’m pretty sure you can identify them.

Bach, Fugue in C-sharp minor from WTC I, BWV 849

We are going full steam ahead, so we have another triple fugue, but this time in 5 voices. By modern speculation, the three themes are now known as the crucifix, the crown of thorns, and the pounding nails. The three themes are explored within the same exposition, so they are very tight-packed in a sense.

I decided to choose a recording on viols instead of on the piano mostly because it sounded a lot more haunting, but maybe you can listen to both and give a comparison.

Bach, Contrapunctus XIV “Fuga a 3 Soggetti” from the Art of Fugue, BWV 1080

Literally, this contrapunctus is called “fugue with three subjects”, and as the name suggests, this is a triple fugue with 3 subjects, but unlike the previous triple fugues, the 3 subjects here are introduced in separate expositions: we have a first fugue with 1 subject, a second fugue with 2 subjects, and then a third fugue with 3 subjects.

The first fugue begins with a slow processional theme, which is used in both its upright and inverted form. Bach, as if teaching all the techniques in counterpoint, allows the voices to enter twice in the exposition, first all in upright form sequentially, and then alternating the inverted and upright forms (first mid-entry 0:48). At the second mid-entry (1:46), Bach demonstrates all of the possible permutations of stretti of two voices, for two upright statements, then two inverted statements, then for one upright and one inverted statement each. At the last entry (2:46), he shows us the stretto of three voices, before wrapping up with a small coda to bring us back to D minor.

The second fugue begins (3:30), in traditional baroque style, with a simple statement of the second subject, which is full of motion as compared to the first subject. After two voices have entered, he teases the listener with the first subject, presented in diminution and with embellishment. After a brief meandering episode, the first mid-entry (4:22) graces us with its presence, entering with the second subject on top, and the first subject as a countersubject. The second mid-entry (4:52) follows soon after, this time with the roles of the two subjects reversed, with the first subject on top. Finally, just like the end of the first fugue, the final entry is presented with three simultaneous statements of the subjects, this time having the second subject and two upright statements of the first subject. Notably, the second subject is presented in the bass register, while the two first subjects are present only in the descant, which allows us to clearly delineate the two themes.

At the exposition of the third fugue (5:32), after 13 full contrapunctii and 4 canons, Bach finally signs his name on the art of fugue, presenting the B-A-C-H motif as our third subject. Just as the first fugue, the first mid-entry (6:01) has the B-A-C-H motif entering in alternating upright and inverted forms, but here, you can also hear the first subject hinted at once in the soprano. In the second mid-entry (6:40), we hear the stretto of all 3 subjects, as Bach teaches us how the 3 subjects is supposed to fit together.

At this point, the manuscript stops, and Bach leaves the art of fugue unfinished. His son, CPE bach, propagates the rumour that Bach died at this precise moment. Every performer at this point has to follow in the master’s footsteps, and compose the ending to a triple fugue. Subsequent entries here present the three subjects inverted, and in stretto, and includes some more modern (chromatic) harmonies, before ending the piece.

Curiously, this is the only contrapunctus out of the set of 14 that does not employ the Art of Fugue theme from Contrapunctus I.

Busoni, Fantasia Contrappuntistica

Our hard work understanding fugues culminates here, with the gargantuan extension to the Fuga a 3 Soggetti that Busoni composed.

Although large in scale, the Fantasia is actually just a prelude and fugue. The prelude itself is an almost faithful transcription of his very own third elegy, written in the form of a theme and variations, which he dubs as a “praeludium chorale”. There are two themes that are featured here, a strong theme A, and a hauntingly beautiful theme B that takes inspiration from a Bach chorale.

After a brief transition, Busoni goes back and trascribes the Fuga a 3 Soggetti. Following its structure, there are 3 fugues: a simple fugue, a double fugue, and then a triple fugue. He applies a slightly more modern flair to these fugues, often making minor changes to double certain voices in octaves for emphasis. Busoni also takes the liberty to extend the third fugue by a massive amount, introducing multiple ways to combine the three subjects.

Then we come to a small break, where we are introduced to a small intermezzo, three fugato variations on the fugal subjects, and then a small cadenza. This leads us all the way back to Contrapunctus I, where he finally states the oft-hinted Art of Fugue subject, and goes into the fourth fugue, the quadruple fugue that forms the centerpiece of the subject.

After the four subjects break down into unrecognizable pieces, Busoni cycles back to the chorale, before thundering out a mutated form of the third subject, the B-A-C-H, and brings the entire piece into conclusion.