WTC I: Fugue No. 1 in C Major, Part III

Starting with WTC I: Fugue No. 1 in C Major, Part I, I began a discussion of the fugue portion of Bach's Prelude and Fugue in C Major from Book I of his Well-Tempered Clavier. I continued the discussion in Part II by talking about the first two of the five cadences Bach uses in this fugue. Now I'd like to extend that discussion by describing the remaining three cadences.

You will get the most out of the discussion if you open this web page in a separate browser window or tab. It contains a Macromedia presentation of the fugue as played by David Korevaar and as analyzed by him and by music scholar Timothy Smith. The presentation includes the score in musical notation, measure by measure, and also a timeline giving an overview of the score.

As I said in the previous post, a cadence is basically a sequence of two chords. The root note of the first chord is typically, but not always, of a higher scale degree than that of the second. Hence, the progression of chords is usually a "falling" one — the word cadence comes from a Latin word for "falling."

For example, C: V - I⁶ is the first cadence in the fugue. It starts on the "FOUR" beat of measure six with the notes G-B-D (the V or "five" chord of the C Major scale; count "one" for C, "two" for D, on up to "five" for G, the root note of the G Major triad). It moves from there to C-E-G-A (the I⁶ or "one sixth" chord, since A, the sixth note in the C Major scale, is added to the basic C-E-G triad that has the keynote of the C Major scale as its root). These notes do not occur all at once, but are spread out, in what amounts to an arpeggio, over all of measure seven.

Prior to this quasi-arpeggio, the notes E and G are heard together on beat "ONE" of the seventh measure, in what can be considered the official completion of the C: V - I⁶ cadence. In a Bach fugue, some notes of the completing chord of a cadence are ofttimes delayed in this fashion.

This C: V - I⁶ cadence thus spans the transition from the sixth to the seventh measure of the fugue, where the exposition section of the music ends and the first development section commences.

The effect of this cadence is to tell the ear:

• what the keynote or tonal center of the music is (specifically, the note C);
• what scale is being used (C Major); and
• what is happening in terms of the structure of the fugue (a new section is beginning)

Because it is a variety of the generic V - I cadence, musicologists call the V - I⁶ cadence an "authentic" cadence — though, as explained in the previous post, this one is an "imperfect authentic cadence" (IAC) and not a "perfect authentic cadence( (PAC). There are other types of cadences, but in this fugue only IACs and PACs are heard.


At the start of the third development section in the timeline, at the transition between measures 18 and 19, we see dm:V - i. Here, the V and i chords of the key of D minor confirm an ongoing modulation into that key.

Two measures later, in the middle of the same section, straddling the bar line between measures 21 and 22, G: V - I betokens a new key of G Major — which happens to be the major key built on the fifth degree of the original "home" key of the fugue, C Major. This is appropriate, because in the run-up to the coda, C: V - I brings us back to that home key. These are the three cadences I'll be discussing in what follows.

I'll get back to dm:V - i and G: V - I presently. For now, I'll skip ahead to C: V - I.

We hear in the tenor, alto, and soprano voices, respectively, the notes D, G, and B at the very beginning of measure 23. Well, the V chord of C Major is ... wait for it ... a G triad made of notes G, B, and D! So when the timeline shows C: V - I at this point, this is the start of it.

You can hear this G-B-D triad by clicking on the timeline at measure 23. Bear in mind that when you do so, the first two notes you hear are, for some reason, the final two sixteenth notes of the previous measure. Only then do you hear the actual G triad — played atop a sixteenth note on F which commences a run of similarly brief notes up and down the scale in the bass voice. This run is a capricious sort of "free association" exercise; it extends a similar one in the soprano voice in the previous measure.

The business of accomplishing the impending V - I cadence, meanwhile, is left to the other three voices: the soprano, the alto, and the tenor.

The rest of measure 23 gives us a musical picture of G-B-D, the V chord of C Major, preparing to morph into the I chord of C Major, C-E-G, at the outset of measure 24. If you look closely at the tenor, alto, and soprano voices, you can see that the notes B, G, and D — the notes which make up the V chord — keep reappearing throughout the remainder of measure 23.

That said, there is also a sort of prefiguration of the resolution of the cadence at beat "THREE" of measure 23, when we hear (from bass to soprano) the notes E, G, E, and C, which describe the upcoming I chord in an inverted variation — i.e., the lowest note is not the root note of the triad, C.

Then, following beat "FOUR," the very last "and" beat of measure 23 has a B in the soprano voice, a D in the alto (which actually began sounding on the "FOUR" beat) and a G in the tenor (which also began on the "FOUR" beat). Meanwhile, the free-form bass run has itself joined in the fun, settling onto a low G note.

At last comes the first notes of measure 24, which (from bass through soprano) are C, G, E, and C: a C Major triad in root position. The G is actually the one that commenced sounding on beat "FOUR" of the prior measure. But the B and D from before are now transformed into C and E. The C: V - I cadence is complete. Bring on the coda!

The coda, the brief section which wraps up this fugue, is also known as the peroration. It consists of measures 24 - 27 of the piece. Through these final four measures the bass voice plays just one long, sustained note, on C. This is called a "pedal" note, and it is no coincidence that the pedal is the keynote of the home key of the fugue. Nor is it a coincidence that the final notes heard in the fugue find C being played, in different octaves, by the bass, tenor, and soprano voices, while the alto voice bifurcates into E and G, so that in addition to the keynote of the home key, C Major, we also hear its most basic three-note chord, the C Major triad, one last satisfying time.


Now, back to the two cadences I skipped over, dm:V - i and G: V - I.

The dm:V - i cadence spans the transition between the second and third development sections of the fugue, lying as it does across the bar line that separates measures 18 and 19. From the "dm" notation that is shown in the timeline, we expect first to find in the score the V chord of the D minor scale, which is the major triad A-C♯-E. Then we expect to find D-F-A, the i chord in the same, D minor scale (written with a lowercase "i" because it is a minor triad).

The easiest way to approach finding this progression as we look at the score itself is to look in measure 18 for a sharp ("♯") sign on one or more notes. Since in the key signature of the fugue there is no sharp sign on C — there are no sharps or flats whatsoever in this fugue's key signature — the C♯'s will have to be notated with "accidental" sharp signs.

Indeed, we find three C♯'s: one in the tenor, and two in the alto. The first of these to be played occurs on the 18th measure's "TWO" beat. It is in the alto, and is accompanied by an E (a held-over note) in the tenor along with an A in the soprano. These three voices are at this juncture combining to play an A Major triad, which is, by no coincidence whatever, the V triad of the D minor scale.

Notice that the bass voice, meanwhile, is doing its own thing. Its note at this point is F, which is not part of the A Major triad. As we saw when we analyzed the C: V - I cadence above, the bass is capable of engaging in a sort of sleight-of-hand diversion from the main business of establishing the beginning of a dm:V - i cadence. But in this case what the bass voice is doing is yet more important: reprising the subject of the fugue.

That bass note, F, is the seventh note to occur in the subject motif ... which may be a bit difficult to detect, in this particular repetition, since the first note is stretched out in length, from its customary eighth note, into a quarter note. I hear this stretching as a sort of hesitancy on the part of the bass voice to launch into the subject motif at all.

At any rate, now that we've found the V chord, A-C♯-E, we're looking next for D-F-A, the i chord of the key of D minor — which, since it is the i chord, is a D minor chord. Not surprisingly, we find it on the "FOUR" beat of the same measure, measure 18. The bass voice has now decided to cooperate with the cadence by playing the A note of this chord, while the tenor voice shuts up briefly. Meanwhile, F and D are played by the alto and soprano voices, respectively.

How quickly things change! On the "and" beat just before the "THREE" in measure 18, we have just heard an A and a C♯, two of the notes from the V chord, along with a sixteenth note on G which migrates hastily to E, the V chord's third note, on the micro-beat between the "and" and the "THREE." In other words, just prior to "THREE" in measure 18 there is another instance of the V chord ... and then, all of a sudden — with the A note on the "and" beat being sustained to become the A note on the "THREE" — the i chord arrives.


But the "THREE" beat of measure 18 is not the actual conclusion of the dm:V - i cadence. That distinction is reserved for the chord that is heard of the "TWO" beat of the next measure, measure 19. There, the notes we hear are, from bass to soprano, D, D, F♯, and D.

Just prior to that chord, on the "and" beat, we hear the V chord again: A-C♯-E. When we hear D-F♯-E on "THREE," we take it to be the i chord at the end of a perfect authentic cadence which began with the V chord. This is so despite two objections.

The first objection is that the note A which, at a perfect fifth interval above the tonic of D, is supposed to be part of this i triad is absent. The is no big problem, because Bach soon has the F♯ which is played as part of the chord ascend, in the alto voice, to A (albeit by way of the intervening G). This happens in the course of having the alto restate the subject of the fugue. Accordingly, the ear hears order restored to the cadential conclusion at the heroic hands of the fugue's own subject — very pleasing, that.

The second objection is that the supposed i chord is not really a minor triad, as it seemingly ought to be if D minor is to be the new temporary key of the fugue. If it were a minor triad, it would be D-F-E, not D-F♯-E. F is a diatonic (scale-based) note in the D minor scale; F♯ is not. What's up with that?

Joseph Kerman's The Art of Fugue

According to Joseph Kerman in his excellent book The Art of Fugue: Bach Fugues for Keyboard 1715-1750, this raising of F to F♯ in measure 19 produces a "novel effect" that

... seems to arise from a compound of the unexpected D-major harmony — with F♯ — the pause on the pedal D [bass note], brief as it is, and the way the two inner voices [i.e., the alto and tenor] ease their way or glide across the cadential downbeat [on "THREE"]. ... The progression conveys something of the mood of an interrupted, or deceptive, cadence, though technically it cannot be called that, of course. (pp. 7-8)

An interrupted or deceptive cadence is one that does not arrive at the I or i chord (depending on whether the scale is major or minor) but winds up on (for instance) the chord based on the sixth degree of the scale. This is not that, but Kerman says it feels like it anyway. He continues:

... to call the D-major chord a tierce de Picardie also seems odd, for Picardy mode-change is supposed to happen to the tonic chord at the end of a composition ...

Here, he's referring to a so-called "Picardy third," a device Bach and others used by means of which to end a minor-key fugue on a major chord by raising the chord's 3rd degree — as F to F♯. That's what he's doing with this chord: raising the 3rd. But it's being done not to the tonic chord here, but to "the dominant of the dominant," says Kerman.

The "dominant" is the 5th degree of a scale. If the scale is C Major, the dominant is G (count: C-D-E-F-G). Then, taking G as the tonic, its dominant — the "dominant of the dominant" — is (G-A-B-C-)D. The triad built on it ought to be (since F and not F♯ is in the original C Major scale) D-F-A. Here, however, Bach exercises the prerogative of his genius to make it D-F♯-A.

So, as the third development section begins, we hear once again that we are — albeit temporarily — in a key other than the home key of the fugue. This time, that temporary key is D minor — despite the F♯. That will change when the G: V - I cadence arrives.


Unlike every one of the cadences I have discussed to this point, G: V - I does not end a section of the fugue. Instead, it is found in the middle of the third development section.

On the "THREE" beat in the 19th measure, which begins in the key of D minor as a result of the dm:V - i cadence just analyzed, we hear (as just mentioned) a note that ordinarily does not belong here: F♯. F♯ is not part of the D minor scale. It is, however, part of the G Major scale. Perhaps its presence here foretells that D minor's time as the reigning temporary key will end quickly, and a V - I cadence initiating G Major as the new temporary key of the piece can be expected soon.

Indeed, a G: V - I cadence is completed (as the timeline indicates) in the middle of measure 21. On the "THREE" beat of that measure, the notes that are heard are G (bass and tenor), D (a held-over note in the alto voice), and B (in the soprano). G-B-D is the I triad in G Major. (Notice that since the soprano voice is playing D and not G, this is an imperfect authentic cadence.)

Given that this event concludes a G: V - I cadence, albeit imperfect, one would expect to find among the immediately preceding notes an explicit V triad of the key of G Major, D-F♯-A, stacked up nicely as a single chord. And so we do. On the "TWO" beat of the same (21st) measure, we have F♯ in the bass, A in the tenor, and D in the alto. (The soprano voice is playing a C, but never mind.)

In the space of just two of the four principal beats of a four-beat measure, a G: V - I imperfect authentic cadence has occurred, and we are now out of the brief temporary key of D minor and into a new temporary key, G Major.

Notice that, although this is technically an authentic cadence, it does not feel like one. There is no sense of conclusion or finality. No sense that a point of articulation has been reached, where one section of the music gives way to the next. At most, we feel like the cadence is just there to limn the change of key from D minor (with a key signature of 1 flat) to G major (1 sharp), en route to quickly settling back into the original key of the fugue, C Major (no flats or sharps). And, in fact, that is exactly what is going on.

So, not surprisingly, the temporary key of G Major will itself hold sway but briefly, until the C: V - I cadence discussed earlier has been completed at the beginning of measure 24.

If you look again at the way that cadence transpires, you will see that G — the "destination" note of G Major — is sounded both in the V chord and then in the I chord of the C: V - I cadence. In the tenor voice, it is even sounded as a single, sustained note that is held over from one chord to the next. The "destination" note of the key that is on its way out (G Major) turns out to be also the "pivot" note by means of which the successor key (C Major) is ushered in.

Clever fellow, that Bach! He is always building harmonic order (chord progressions) out of supposedly "free" melodies (voices) by using such devices as pivot notes and faux Picardy thirds. That is, after all, the essence of the art of the fugue.