I/6.1.1 The prelude-type

This prelude sets out from a purely harmonic design. Against this backdrop, melodic ideas gradually emerge, in some cases emancipating themselves from simple bass patterns. The piece should therefore be analyzed under the criteria of both harmony and motivic development.

I/6.1.2 The design of the prelude

The first cadence can be detected at the end of the lower D which serves as a tonic pedal during the first five beats of the piece. (In the right hand, the conclusion of this initial cadence falls on the fifth triplet-sixteenth-note of bar 2). The fact, however, that this progression unfolds and closes above a pedal-note bass conveys the impression that the piece has not quite begun yet. This cadence-ending is thus not to be regarded as a structural break, but as a first segment of a larger unit.

The more encompassing first section of the prelude modulates to the relative key, F major. Its cadential close appears in bar 6 (right hand: until the eighth triplet-sixteenth-note). The structural caesura is enhanced by a distinct cadential-bass pattern (see bars 5/6).

Seen under harmonic aspects alone, the sections of this prelude are as follows:

  

	I	bars 1 - 6	tonic to relative	(D minor to F major)
	II	bars 6 - 8	relative to subdominant	(F major to G minor)
	III	bars 8 - 10	subdominant to dominant	(G minor to A minor)
	IV	bars 10 - 15	dominant back to tonic	(A minor to D minor)
	V	bars 15 - 26	tonic confirmed	(D minor)

However, a closer look at the melodic structure reveals that the three sections in the middle are closely interrelated: the first sets up a motive in the lower voice which is sequenced one tone higher immediately after its introduction (compare L: bars 6-8 with bars 8-10) and taken up again after a short interruption (see L: bars 12-14). This time, the expected cadence is delayed and appears only after an additional bar of extension (see bar 15). Furthermore, what was just passed by as an "interruption" contains in itself a short motive (see bar 10: A A G F) which is sequenced three times.

Extending the analytical view to include motivic development, the previous layout can thus be condensed into three major sections:
  

	I	bars 1 - 6	(D - F)
	II	bars 6 - 15	(F - D, two motives and their development)
	III	bars 15 - 26	(D - D, motives 1 and 2 not present)

The sequential structure in the three harmonically closed progressions in section II apart, the prelude does not contain any transpositions, variations or other analogies of passages. However, a very short correspondence deserves a mention as it will add to the understanding of the overall design: bar 15 recurs in bar 23, transposed to the dominant and enriched by an additional middle voice.

I/6.1.3 Practical considerations for performers

The continuous flow of broken chord patterns in the right hand might indicate an ideal tempo for the piece. As in those preludes which are determined solely by harmonic processes, these broken chords should be played fast enough to allow listeners to perceive them as harmonic entities rather than as melodic zigzag lines. On the other hand, they should not sound like sparkling virtuoso figures so as not to divert the listener from grasping the inherent harmonic design.

The appropriate articulation in this prelude should reflect the rather lively basic character: legato for the triplet-sixteenth-notes but non legato for the eighth-notes. Within this non legato articulation in the eighth-notes, a color shading between neutral bass notes and more emotional motivic notes is, of course, very much encouraged.

I/6.1.4 What is happening in this prelude?

The first larger section of this D minor prelude, stretching from bar 1 to bar 6, displays all the characteristics of a piece determined by harmonic processes. Each eighth-note represents a complete chord, and unlike the more complex progressions in the preludes in C major, C minor, D major etc., most of these chords (until the third eighth-note in bar 4) appear in root position.

Furthermore, both of the secondary processes which often occur in harmonically-determined pieces can be found here: a pedal in bars 1/2 and sequences in bars 2/3, 3/4 and 4/5. These processes are connected with the following dynamic lines:

 

-	Above the tonic pedal, a D minor cadence unfolds with the active step (i-iv) taking place on the middle beat of bar 1. This tension-enhancing active step is reinforced by an upward shift in the pitch level. A complementing descent accompanies the tension-decrease towards the resolution in bar 2:   bars 1 - 2(fifth sixteenth-note) p - mp - p
-	After a drastic leap of an octave which brings a sudden tension-increase, a model of four eighth-notes is introduced and sequenced one step lower. Within the model, the tension as well as the pitches are again descending:   bars 2 (beat 2) - 3 (beat 1) mf - - p+ - mp+ - p-
-	Within the next bar, the density of both the sequence pattern and the harmonic change doubles. Each eighth-note now establishes a new chord, and the model which is to be sequenced is only two eighth-notes long. (The sequences contain one slight adaptation in the bass which allows the left hand to move to the preferred lower octave.) This model and its three sequences move upward and thus express a rather significant tension-increase: bars 3 (beat 2) - 4 (beat 1) mp - mp+ - mf - - mf
-	The next bar brings a yet slightly different pattern, with harmonic changes now occurring only on the quarter-note beats while the bass inserts non-harmonic passing notes on each unaccented eighth-note. The two-eighth-note long model is followed by three descending sequences. Here, the tension-release is prolonged for an additional bar with a short cadential close: bars 4 (beat 2) - 6 (beat 2) mf + -------------- p

The second section of the prelude is determined by its two motives. The larger first motive (M1), introduced as an independent melodic unit in the lower-voice line of bars 6-8, is actually not completely new. Its stepwise descent recalls the bass line in bar 4, and its ending is clearly reminiscent of the cadential-bass pattern in bars 5/6. Some details, however, are new, and they are very effective:
 

-	M1 begins on a pitch which is the seventh of its chord – this creates high melodic tension right from the start of the motive.
-	In the second half of the motive the line returns once more to the same note which is now redefined as the root of the deceptive chord of the cadence which follows – the tension created here is of a harmonic nature.

 
In the right hand, the uppermost chord notes present several short parallels to the left-hand line. Particularly the ascent in the middle of bar 7 serves to enhance the impact of the deceptive chord.

The much shorter second motive (M2), introduced in bar 10 (second to fifth eighth-note) depicts a melodic descent combined with a harmonic process of relaxation; it thus counterbalances the gesture of M1.

The dynamic processes in this section may be described as follows:
 

-	motive 1	=	bars 6 (beat 2) – 8 (beat 2) mf - mp - mf + - p
-	sequence	=	bars 8 (beat 2) – 10 (beat 1) mf + - mp+ - poco f – - p+
-	motive 2 + sequences	=	bars 10 (beat 1) – 12 (beat 2) mf - mp+ - mp- - p+
-	motive 1 + extension	=	bars 12 (beat 2) – 15 (beat 1) poco f -- mf - - poco f - mp – - mf + - p

The third section of the prelude combines secondary melodic processes of a different nature. The lower part sets out from the newly-found keynote D. After a broken chord which propels it almost two octaves up, it splits into a hidden two-part structure, with D as a pedal in the lower part and a melodic descent in the upper notes (see the falling line from a major seventh to a second interval in bars 16-19). The right-hand part accompanies this descent first with a peak-note figure which contracts as the bass interval shrinks. The anticipated complete annihilation of tension is delayed when the peak-note figure takes on a virtuoso form (bars 18/19); but, as this virtuosity seems to bear little effect against the decreasing power of the bass line, the previous peak-note pattern is reinstated.

Soon after the bass descent is concluded, the concealed two-part structure of the left-hand part turns into a real one: an independent middle voice evolves. As middle and lower voices proceed, two distinct tritone intervals mark points of high tension (see the middle beats in bars 21 and 22). In the right-hand part, the peak note figure is taken over alternately by the top and bottom notes of the broken chords. The overall impression is one of descent, so that the downbeat of bar 23 is reached in a softness almost comparable to that of its corresponding bar 15. From here, both hands are once again propelled up almost two octaves, with the now firmly established middle voice adding extra drive. In an unaccompanied treble line the ascent continues until the high B>ð, from where diminished chords descend chromatically until the keynote D is regained – and with it the beginning of a short, chordal-style final cadence.

I/6.2.1 The subject

Beginning after one eighth-note of rest and ending on the downbeat of bar 3, this subject is exactly two bars long. Despite the sudden interruption of the sound flow indicated by the wedge on the B^b which creates a tension-sustaining hiatus, the subject must be interpreted as consisting of one indivisible phrase.

The pitch pattern in the subject contains predominantly steps and minor thirds. However, there is one larger leap in the middle of the phrase. This leap embraces a minor sixth and thus represents one of the "high-tension intervals". The rhythmic pattern contains four different note values: sixteenth-notes, eighth-notes, quarter-notes and the tied quarter-note (which, as will be seen later, does not belong entirely to the subject but is nevertheless heard as such a rhythmic unit).

As one compares the remainder of the piece rhythmically with the subject if becomes obvious that only one additional note value appears within the thematically active parts (i.e. not counting the longer values in the cadential closes): a syncopated dotted-eighth-note note. It is the same as that sounding implicitly at the beginning of the counter-subject.

The harmonic background to the subject is that of a simple cadential progression; the tonic gives way to the subdominant in bar 2 beat 2, followed by the dominant seventh in bar 2 beat 3, and resolving onto the tonic on the downbeat of bar 3.

(ex. 55)

 

The fact that the active harmonic step takes place, in the middle of bar 2 and not on one of the downbeats, endows the B^b with the quality of a syncopation. This impression is enhanced by the rhythmic standstill (this is the feeling given by the quarter-note which appears suddenly, after the initial eighth-note motion had developed into sixteenth-notes) and the interrupting wedge.

This B^b is therefore the obvious choice for a climax. Melodically reached in a high-tension leap, harmonically representing chord iv, further enhanced both by its suddenly larger rhythmic value and by the unexpected articulation mark, and metrically appearing as a quasi-syncopation – this note certainly combines all possible features which could characterize a climax.

While it seems thus perfectly clear where the climax is, both the preparation and the resolution of this climax are slightly irregular. On the one hand, the process of tension-growth before the climax is impeded by a slur in bar two. (For more details regarding the way in which this slur can be interpreted, see paragraph 7 below.) On the other hand, the climax is followed by only two notes which have to cope with the task of resolving the rather powerful tension. Yet, instead of ending on either of the melodically relaxed notes of the tonic chord, the third F or the keynote D, the subject comes to a melodically incomplete-sounding halt on the fifth A – the one scale degree which regularly fails to convey a feeling of resolution.

I/6.2.2 The statements of the subject

The subject appears in seventeen complete and seven incomplete but structurally relevant statements (the latter are here marked with an asterisk):

(ex. 56)

Two kinds of changes can be observed in the complete statements of the subject: Its ending can be slightly varied (see bars 22/23 with a fifth interval, bars 35/36 with eighth-notes instead of the two quarter-notes), or it can appear inverted (see entries nos. 5, 7, 12, 13, 14, 15, 16, 18, 22).

Furthermore, among the complete subject statements there are two conspicuous two-part strettos, both involving complete entries appearing in the original shape (see in bars 17-20, 39-42). These are surpassed by a stretto which sounds in a chain of three entries of which two are inverted (see in bars 21-25). In addition, there are several strettos made up of a mixture of complete and incomplete statements, using both the original shape and the inversion:

-	bars 12-16:		incomplete/inverted (M)	+	complete/original (U)
		+	incomplete/original (L)	+	complete/inverted (M)
-	bars 25-31:		incomplete/inverted (M)	+	incomplete/inverted (L)
		+	complete/inverted (U)	+	complete/original (M)
				+	complete/inverted (L)
-	bars 33-36:		incomplete/original (M)	+	complete/original (Lvar)
		+	incomplete/original (U)	+	incomplete/inverted (M)

Parallels of subject statements do not occur. There are, however, two instances where the (inverted) second entry of a stretto coincides with an incomplete entry in the original shape: see

bars 14/15 M + L,

bars 35/36 U + M.

I/6.2.3 The counter-subject

Bach uses only one counter-subject in his D minor fugue. CS begins immediately after the first subject statement. Here the fact that the subject ends in a tied note brings up the question of what note value the beginning of the counter-subject is worth.

To what proportion the A which is held for the value of five sixteenth-notes on the downbeat of bar 3 is divided between the subject and its counter-subject can be understood when comparing later statements of the counter-subject: whenever CS does not follow immediately after a subject statement, it invariably begins with a first note of dotted-eighth-note duration (see e.g. bar 6 upper voice, bar 15 upper voice, varied).

The counter-subject thus begins, like the subject, on the second eighth-note of the bar and ends after two bars on the downbeat (with varying note values for its final note). It consists of two subphrases which are interrelated through their rhythmic structure. Both subphrases set out with a syncopated dotted-eighth-note note on the second eighth-note of the bar (compare U: bars 3 and 4) and both are followed by regular sixteenth-note motion until the subsequent downbeat. Moreover, in each subphrase the sixteenth-note motion presents a sequential pattern: in the first subphrase, four descending sixteenth-notes are repeated one step lower; in the second subphrase, a "turn" figure is equally repeated one step lower. (The two subphrases are particularly clearly distinguished in bars 6-8 where the first is allocated to the upper voice while the second continues in the middle voice.)

Considering all these features, each of the subphrases will find its climax on the initial syncopation, after which the sixteenth-notes in their descending direction bring a relaxation. (These two dynamic developments are very gentle.) With regard to the relationship between the two subphrases, it seems that the second subphrase represents a higher tension-level than the first one, both because of the shift upwards and, more particularly, because of the greater emphasis created by the "real" syncopation as compared to the "implied" one at the beginning of CS.

The sketch shows the phrase structure and the dynamic design in subject and counter-subject (ex. 57).

 

 I/6.2.4 The episodes

Determining the number of episodes in the D minor fugue depends on the evaluation of the incomplete subject statements. If one assumes that, as in many other fugues, considerably shortened statements of the subject function as episode material, then the number of episodes would amount to eight. However, since the incomplete statements in this particular piece mostly engage in primary-level combinations (strettos and parallels) with complete entries, the label "episode" seems ill-fitted for those bars which are characterized by these shortened entries.

The following table lists the episodes of this fugue but places those bars which contain incomplete subject statements in brackets:

Almost all episode material in this fugue seems closely related to the subject. This is particularly true for the first three episodes and their analogues later in the piece:

In E1	the upper voice imitates the second half of the subject (see bar 5 r.h.) while the middle voice recalls the beginning of CS – including the way in which CS emerged out of the subject's ending note (compare M: bar 5 with U: bar 3).
In E2	both the upper and lower voices extend the preceding entry in sequences (see bars 9-10d sequenced in bars 10-11d and 11-12d); only the middle voice gains some independence by creating a little motivic figure of one-bar length. Bars 12-13d then combine the first incomplete entry in the middle voice with the second half of the subject in the lower voice (as it appeared in E1) and a long note in the upper voice.
E3	is a varied sequence of the bar preceding it (compare bars 16-17d with bars 15-16d).
E4,	by contrast, is a typical cadential close: the upper voice provides an elaborately ornamented keynote / leading-note / keynote (do-si-do) formula, the middle voice contributes the syncopated trill with resolution on the next downbeat which often appears in closing-formulas, and the lower voice sounds a cadential-bass pattern. In E5, two incomplete subject entries in the middle and lower voices are combined with a figure which derives from the end of the counter-subject – extended and in inversion.

The three final episodes are related to three episodes in the first half of the piece:

E6 see E2	although the voices are inverted and considerably varied (compare bars 31-33d or 34d with bars 10-12d or 13d).
E7 see E3	the latter being an extended version of the former, but its three bars equally continuing as a sequence of the preceding bar.
E8 see E4	insofar as the first bar of E8 is an exact transposition of E4 (compare bars 42-43d with bars 20-21d). This is followed by one and a half bars of coda in which the three voices of the fugue are split into six. What are now the four inner voices present a fourfold parallel of the first half of the subject – the right-hand double thirds in inversion, those of the left-hand in the original direction.

 
Among these eight episodes there is not a single one which displays independent motives, and none reaches the level of a self-contained unit. One can, however, distinguish four different ways in which these episodes behave within the dynamic design of the fugue:

-	The two cadential closes (E4 and the first half of E8) create the most noticeable relaxation. In the case of E8, this relaxation is soon counteracted by a particularly powerful tension-increase, but this represents already the coda.
-	Those episodes which appear as an extension of the previous development by picking up parts of the preceding material in imitation or sequence, create a gradual relaxation but never quite lose the color of the primary material (see E1, E2 and E6).
-	Those episodes which extend the preceding bar in sequences involving all voices (E3 and E7) serve to prolong the tension of the subject entry; in both cases this impression is further enhanced by the incomplete statements which balance the resolving tendency in the second half of the subject (see L: bars 15-16d and particularly U: bars 35-36d).
-	Finally, that episode which appeared in brackets because of its stretto of incomplete subject statements (E5) has the strongest drive of all episodes. It sets out from a point of complete relaxation and creates, in both its incomplete stretto and the inverted (i.e. ascending) counter-subject figure, a powerful increase which prepares the subsequent complete-entry stretto.

I/6.2.5 Character, tempo, articulation, ornament realization

Both the pitch pattern (which shows predominantly steps and minor thirds, interrupted only by the high-tension interval of the minor sixth in the subject) and the rhythm (which includes five different note values) advocate a rather calm basic character. Yet there is one detail which seems to support a character in which the single notes carry less weight: it is the fact that, upon close inspection, all sixteenth-note groups are either turn-figures or scale segments which ornament larger notes (e.g. bar 3 = A-G ornamented).

Taking into consideration both the main constituents of the basic character and the ornamental structure of the sixteenth-notes, the ideal tempo is one which creates a sense of calm within a rather fluent quarter-note pulse. The corresponding articulation requires legato for all melodic notes, but giving slightly less weight to the sixteenth-notes. Non legato is reserved for cadential-bass notes (see bars 20 and 42) and obvious broken chord patterns (as in L: bars 31/32). The wedged note in the subject should sound actively interrupted after about half its note value. (For those who like to work on shades: such active interruption stands in contrast to the passive ending of unmarked notes before a rest or phrase cut).

Special discussion is needed for the slur in the subject. What does this slur indicate? As we all know that articulation symbols in Baroque polyphonic music derive from string and wind players, it might help to take a string player's view. A violinist who approaches this piece would play each of the notes with a separate bow movement, while attempting the best legato due to the rather calm basic character which he recognizes in this fugue. He would thus be able to increase the tension through every note. But, seeing the slur in the subject, he would change his attitude and combine the group of notes comprised under the slur in one single bow movement. The dynamic effect is that the notes following the initial note under the slur, sound more passive and do not continue the tension. The slur thus creates a two-level structure: the dynamic gesture leads through each of the notes in the first half of the subject to the first note under the slur and from there directly to the climax note B^b, after which it breaks off and then resolves through the trill into the ending note of the subject.

The relative tempo of the prelude to the fugue is simple:

For the ornament in the subject there are two possible solutions; they depend on how the individual interpreter feels about the wedge on the climax note.

-	For performers who feel that the wedge creates a sudden, rather dramatic halt in the melodic flow, and that the G which follows comes in after something like a phrasing, the trill should begin according to the rules for ornaments at phrase beginnings – i.e. on the main note.
-	For performers who feel that the wedge creates articulation rather than phrasing, and that the tension is suspended throughout this interruption and picked up at almost undiminished level in the G, the trill should begin according to the rules for ornaments within a phrase – i.e. on the upper auxiliary note.

In each case the trill shakes in thirty-second-notes and ends in the suffix F-G.

I/6.2.6 The design of the fugue

There are several indicators which help determine the design of this fugue:

-	A subject entry in reduced ensemble can be found at the beginning of the four-bar passage with multiple stretto (see bar 21 beat 2 to bar 25d). Moreover, this reduction of voices occurs after that episode which was identified as a definite cadence with strongly conclusive powers.
-	The fugue reveals a rather extended analogy. Working backwards from the corresponding cadences one can find: bars 20-21d correspond with bars 42-43d bars 17-19d correspond with bars 39-41d bars 12-(16d) correspond with bars 33-(38d) bars 9-11d correspond with bars 30-32d
-	The order of statements in the fugue's first half shows several distinct patterns: * Bars 1-10 feature four single entries in the voices U, M, L, U. * Bars 13-16 present, after a preparation by an incomplete entry, a stretto with two complete and one incomplete subject statements (complete entries in voices UM; for more details refer back to the table of statements). * This complex stretto is followed in bars 17-20 by a straightforward stretto of two complete (uninverted) entries in voices LM. The fugue's second half begins differently but then shows a similar design: * Bars 21-25 feature three statements in stretto chain – a leading entry in the lower voice accompanied one bar later by the inverted entry in the upper voice and followed by another inverted entry again in the lower voice. * Bars 27-31 follow with another complex stretto built, after two preparing incomplete statements in bars 25/26, by three complete entries in voices U, M, L. * The ensuing complex stretto in bars 34-36 (with preparation in 33) and the straightforward two-part stretto in bars 39-42 correspond with the above-mentioned stretto groups in the first half of the fugue; complete entries are heard in voices L and LM.

The design which can easily be deduced from these findings reveals that the first four simple entries are matched in the second half of the fugue by two strettos. These portions make up the first and third sections, with section endings falling on the downbeats of bars 13 and 34 respectively. Those two long stretches which were found corresponding – bars 9-21 and 30-43 – provide the second and fourth sections.

It is interesting to detect that the harmonic endings of the first and third sections overlap with the incomplete first entries of the strettos with which the second and fourth sections begin. Such an overlapping creates a particularly tight-knit linkage between two consecutive sections, so that the main impression of the design of this fugue is one of two parts, each with a section-pair.

With regard to the harmonic layout, the first three subject statements are firmly rooted in the key of D minor, while the fourth entry seems to belong to no definite key area. The ensuing episode modulates to the dominant key (A major). The second section remains in the dominant region, changing only between the major and minor modes, and closes with another A minor cadence. In the second half of the piece, the very first stretto modulates back to D minor which is reached on the downbeat of bar 25. As with the dominant key in the first half, the entries which follow in the second half change between the major and minor modes. The cadence which corresponds with that of the first half closes still in D minor, but the coda reverses this through the traditional ending of minor-mode pieces in the major mode (Picardy third). For a sketch showing the design of the fugue in D minor, see ex. 58.

I/6.2.7 The development of tension

Within the first section, the tension rises gradually owing to the growing number of voices. The descending sequences in the episode which ends this section serve to slightly dissipate this tension, so that the second section sets out from a rather soft level. Its many strettos, however, create a powerful buildup until immediately before the ending cadence. Both sections feature an episode in their middle which, as has been shown above, is so closely related to the primary material that the tension-loss is negligible.

The third section begins in reduced ensemble on a soft level into which it falls back in the return modulation (bar 25) after only little increase in-between. Yet from here on, all forces join in a stretto which reaches an intensity considerably above that in the corresponding bars of the first part.

The fourth section begins again slightly softer than the third section had ended – but louder than its equivalent in bar 13. After the initial stretto, the tension-sustaining episode with its ascending sequences is extended (compare E7 with E3) and thus leads to a final statement pair which also outranks the corresponding one in the first half of the piece.

It can thus rightly be said that

-	each of the two parts of the fugue is composed as one long buildup of tension which is only slightly interrupted at the section ending in its middle;
-	the second part of the fugue combines several additional intensifying features and thus results, in terms of tension, as a heightened variation of the first part.