Binary Form in Bach's Fugue in D minor, WTC I
Jennifer Taylor


William Rothstein’s idea of inner and outer form provides useful insights into the form of Bach’s Fugue in D minor, WTC I.  For Rothstein, inner form is the tonal or harmonic plan of a work, whereas outer form is the layout or presentation of thematic or motivic ideas.1  When inner and outer forms are identified, it is often questioned whether a relationship exists between the two.  In this particular fugue an inner and outer form are present, but a relationship between the two is not so easily identified.  This is the first fugue in the WTC to employ binary form.  It is clearly articulated and provides the fugue with a harmonic plan.  Beyond the exposition, motivic material is presented, and development is achieved, through the construction of six points of imitation.  These six points of imitation are saturated with the motivic material found in the exposition and afford the listener little relief from the subject.  What becomes clear is that while the outer form develops the fugue, the inner form provides a sense of organization and balance through its clearly articulated form.  This paper will address the inner and outer form of this fugue by discussing the fundamental structure, motivic saturation and points of imitation.



Fundamental Structure

Heinrich Schenker’s views on fugal form, as outlined in section 6 of Free Composition, offer a useful starting point in examining the form of this fugue in D minor.

“Despite the fact that each one exhibits a different design, the fugues of J.S. Bach are genuine fugues in the strictest sense; they are always determined by the subject, by its dimensions and harmonic content, and are controlled by a fundamental structure.”2

The control of the fundamental structure Schenker refers to is important.  Fundamental structure, or Ursatz, refers to a voice-leading structure that controls the tonal and harmonic aspects of the fugue.  Schenker finds that this control is established by the fifth-relation of the first three entries, I - V - I.  The fifth-relation provides the form with direction and, as a result, penetrates the overall form. Therefore, the fugue should show a harmonic movement of I - V - I.


Schenker’s analysis of the D minor fugue demonstrates the principle of the fifth-relation.  The subject enters in m. 1 in the tonic key, is followed by an answer in the dominant key in m. 3, and the third statement of the subject returns to the tonic key in m. 5.  This allows the fundamental structure to gain control of the fugue.  Schenker finds that mm. 1-12 are in D minor, mm. 13-27 are in A minor, and mm. 28-42 have returned to D minor.3  A discussion of motivic saturation will reveal further insights into the fundamental structure Schenker has suggested and its relation to binary form.



Motivic Saturation


Beyond the fundamental structure this fugue is saturated with the motivic material found in the exposition.  Professor Tim Smith refers to the extensive use of the subject and accompanying motives as an “explosion of idea,” and states, “better than 90 percent of this fugue emanates from its first four measures.4 The first four measures of this fugue belong to a formal exposition delineated by a half cadence occurring in mm. 8-9.  Within the first four measures the subject is
stated and, rather than a countersubject, motives "a" and "b" are presented. This is illustrated in Example 1.


Ex. 1


A close look at the motivic content of this fugue reveals that Smith is correct in his statement.  Important to note are those single measures devoid of full subject statements: measures 12, 16, 25 and 26. Despite lacking complete statements, they do incorporate either a portion of the subject or one of the motives.  These measures are illustrated in Example 2.


Ex. 2




Episodes are passages free from statements of the subject.  Although the episodes in this fugue correspond with the definition, they are based on the subject and accompanying motives.  An altered subject statement that begins in m.8 precedes the first episode, mm. 9-12.  This allows the tail of the subject to be used in the upper voice of the episode while motive "b" is heard in the lower voice.  The second episode, mm. 30-33, is a recomposition of the first episode.  In this case the voices have exchanged material.  The third episode, mm. 36-38, utilizes the subject head in the lower voices and motive "a" in the upper voice.  These episodes do not provide a true sense of relief from the subject, as it is only relief from a full statement.  They are instead saturated with subject material and motives "a" and "b", providing the fugue with a sense of unity.


Development in this fugue does not refer to a formal division but rather a compositional concept.  Following the exposition and first episode, rather than introduce new motivic material, Bach utilizes combinations of the subject in its original and inverted form in stretto, forming points of imitation. Motives "a" and "b" are also heard throughout the piece.  These motives appear in each instance of stretto and, with one notable exception that will be discussed later, are not inverted. 


The most significant insight that the discussion of motivic saturation brings to light is that there are only two measures in this fugue that do not appear to derive

their motivic content from the subject or motives "a" and "b": mm. 20 and 42.  These measures are identical in content and form perfect authentic cadences, differing only in key as m. 42 is transposed up a perfect fourth due to the return to D minor.  These significant cadences divide the fugue into two sections, therefore, delineating a binary form.  Measures 20-21 and 42-43 are illustrated in Figure 1.


Fig. 1




The binary form of this fugue seems to contradict Schenker’s concept of fundamental structure that suggests a three-part form.  It is possible to reconcile the two plans, however.  Rather than thinking in terms of I – V – I, Schenker’s voice-leading can be understood as I – V / V – I.  That is, the first section begins in the tonic key and ends in the dominant key, V.  The second section begins in the dominant key, V, and returns to the tonic key, therefore coinciding with the binary form, mm. 1-20, and mm. 21-42.  This reconciliation clearly articulates an inner form as the binary form provides a harmonic plan.


The saturation of motivic material has other implications on the form of this fugue.  First, with such a high degree of saturation an expansive form would exhaust the available material. Secondly, in developing the subject through use of inversion and not introducing new material, a lengthy fugue would risk becoming repetitive and predictable for the listener.  As a result, this fugue is compact containing only 44 measures.  Lastly, with little relief from the subject, the binary form provides the necessary organization and balance.



Points of Imitation


The points of imitation which occur after the exposition utilize the subject in its original and inverted form, accompanied by motives "a" and "b" in stretto.  While the prolific use of the subject and accompanying motives help to unify the fugue, the different voice combinations, increasingly complex stretto and changes in register help to achieve development.  The first two points of imitation beyond the exposition occur in the first section and focus on the original form of the subject.  The remaining four points of imitation occur in the second section and primarily use the inverted form of the subject, serving to develop the fugue.  Although these points of imitation saturate the fugue with the subject and provide unity, balance and organization emerge as a result of the binary form.  The exposition and six points of imitation are illustrated in Example 3 in relation to the binary form.


Ex. 3




The first point of imitation after the exposition occurs in m. 13.  The subject is stated in the soprano and the alto follows in stretto stating the inverted subject.  All three voices are heard during this point of imitation, allowing the low, middle and high registers to be heard.  In the m. 16 a Bb appears in the soprano.  This is a significant note as it is the highest in the fugue.  This Bb serves as an upper neighbour to the fifth scale degree in the fundamental line and is heard in three other important areas that will be discussed later.5


Although the music is still in D minor, this point of imitation is stated at the dominant level.  The importance of this point of imitation is that it presents something new: inversion and stretto.  The motivic material of the exposition is used in a new way demonstrating how this fugue will later be developed.  Supporting this point of imitation is a rising chromatic bass line beginning with the E in m. 14 and concluding in m. 17 on the A, producing an imperfect cadence.  This rising chromatic progression can be likened to the subject and its rising character.  Therefore it extends the point of imitation one measure further than where the alto had concluded its inverted subject statement.  The first point of imitation is illustrated in Figure 2.


Fig. 2


The second point of imitation incorporates the alto and bass voice.  This change in voice combination allows the register to descend.   Although all three voices are heard again in this point of imitation, the soprano descends from the previous measures producing an emphasis on the middle register. 


In binary form the first section often modulates to the dominant key or a closely related key.  The original form of the subject is stated in m. 17 by the bass.  It is preceded by the imperfect cadence at m. 16 producing a modulation to the dominant key of A minor.   Having established the dominant key, these final four measures of the first section conclude with the perfect authentic cadence in m. 20 that delineates the binary form.  A connection to the inner form can be found in mm. 17-20.  These measures are repeated in mm. 39-42 in the key of D minor, forming the final point of imitation.  This is what Schenker refers to as a transfer of fundamental structure.6  The final point of imitation shows a harmonic motion of I-V-I while the fundamental line descends from the third scale degree in m. 39 to the E in m. 42 and concludes on the D in m. 43.  The same harmonic motion and descent of fundamental line is found mm. 17-20 in the key of A minor.  The third scale degree in A minor, C, is presented in m. 17, descends to B in m. 20 and concludes in m. 21 on A.  An excerpt from Schenker’s analysis of this fugue7 is provided below to illustrate the transfer of fundamental structure.


Fig. 3


These first two points of imitation provide unity through the use of the subject and also help develop the fugue by changing the voice combinations and register.  The binary form provides organization, as the first section focuses on the original subject, introduces the idea of inversion and stretto, produces the necessary modulation to the dominant and concludes with the perfect authentic cadence that delineates the binary form.


The second section focuses on the inverted form of the subject beginning with the third point of imitation.  Measure 21, the beginning of this point of imitation, appears to be a recomposition of m. 3, the lower octave and C# in the bass being the only changes.  It is interesting that the bass voice would begin this point of imitation having just begun the last one.  However, because of the perfect authentic cadence in mm. 20-21 the bass entry cannot be inverted without changing the root position A minor chord to a second inversion chord.  To begin with the subject in its original form in the soprano would be redundant as it was already heard in m. 13.  Beginning with the bass entry reinforces the dominant key and also helps to draw attention to the first inverted statement of the subject in the soprano.  The first inverted statement in the bass follows this.  One final point to consider in regards to this point of imitation is that although it is a three-part stretto, it does not involve three voices.  Beginning with all three voices would not leave many compositional options for the remainder of the piece.  Therefore, Bach involves the soprano and bass as the alto has already had an inverted statement of the subject in m. 14.


Serving as a link to the fourth point of imitation is mm. 25-26.  While the alto and bass enter in stretto to one another with inverted statements of the subject head, the soprano has the only inverted statement of motives "a" and "b".  Beginning with the first two beats of motive "a" in inversion, motive "b" then takes over until the end of m. 26.  This notable instance of inversion shifts the register back up in anticipation of the major point of imitation that begins in m. 27.


Framed by inverted subject statements in the soprano and bass, the alto has the first statement of the subject in its original form and key since the exposition in m.28.  Although the return to d minor began in m. 22, it is at this point of imitation that d minor is established, corresponding with the voice leading of the fundamental structure Schenker articulated.  The return to the tonic key seems somewhat early considering it occurs not even halfway through the second section.  Bach most likely returned to the tonic key at this point because the second and third points of imitation were heard in the dominant key and a change was needed.


The alto statement in m. 28 is also accentuated by the second appearance of the highest note, Bb in the soprano, acting as an upper neighbour.  A parallel between this point of imitation and the exposition can also be drawn.  Following the order of entries presented in the exposition, the fourth point of imitation is the only three-voice stretto incorporating complete subject statements. Taking this into consideration, this point of imitation can be viewed as an exposition of the inverted subject and is illustrated below in Figure 4. 


Fig. 4


The fifth point of imitation produces the climax of the fugue.  Here Bach combines both inversion and stretto in a complex manner.  All three voices are included, but a second statement by the alto in m. 35 produces a four-part stretto.  Beginning in m. 33 with the alto, the bass follows and in m. 35 the soprano and alto sound simultaneously.  While the bass has the longest statement, incorporating the subject head and following sixteenth note figure, the alto and soprano utilize only the subject head.  The thickness of texture, close range of voices and build-up of eighth note motion produces the climax. The episode that follows, mm. 36-38, builds upon this tension with a rising chromatic progression in the bass, which can be viewed as a development of the rising character of the subject.  The bass voice also contains leaps of a 9th and 11th in mm. 37-38 that increase the tension. The third appearance of the highest note, Bb, also occurs during this episode in the soprano in m. 38, which again serves as an upper neighbour.  The fifth point of imitation and the following episode is illustrated below in Figure 5.  


Fig. 5


When binary form is employed, the ending of the second section is a transposition or recomposition of the first section’s ending.  The final point of imitation, mm. 39-42, is a transposition of the first section’s ending up a perfect fourth.  Here the binary form, or inner form, is clearly controlling the presentation of the outer form.  What is interesting is that typically the ending would be transposed down a perfect fifth.  Musically this would not have worked because of the register of the previous episode and the fact that a transposition down would detract from the thickness of the coda, mm. 43-44.  To facilitate the transposition of a fifth, the entire second section would have to be transposed up, putting the register far too high, which would lack musical sense.  It is clear that the inner form of the fugue controls this final point of imitation.


The transposition of the first section’s ending typically creates symmetry between each half and, in some cases, the second half can be understood as a recomposition of the first section.  The characteristic transposition is present in this fugue.  As a result two balanced sections emerge that provide a sense of balance and organization amongst the motivically saturated quality of the points of imitation. This is illustrated in Example 4.


Ex. 4


The relationships between section 1 and 2 produced by the binary form also create a balance in the production of tension.  Although the second section has a developmental character and contains the climax of the fugue, symmetry is evident between the two in terms of tension.  As illustrated in Example 3, mm. 1-12 and mm. 21-33 are related in terms of size and content.  Also important to note is that both sections establish the key of D minor and contain episodes that move away from the tonic key, producing tension as the fugue journeys away from home.  It is the first and fifth points of imitation, beyond the exposition, that are most significant in terms of tension.  Mm. 13-16 introduce inversion and stretto and, as is typical in most binary forms, begin to move away from the tonic key producing the first significant rise in tension.  The fifth point of imitation moves away from D minor to G minor and involves the most complex use of inversion and stretto, producing the climax of the fugue.  The first point of imitation occurs 13 measures into the first section and the fifth point of imitation occurs 12 measures into the second section, thus producing symmetry and balance between the sections in terms of tension. The transposition of endings characteristic of binary form also creates symmetry in the resolution of tension.   As stated earlier, the sixth point of imitation is a transposition of the second.  These points of imitation conclude each section and also resolve the tension of the previous points of imitation.  Although the second section is more developmental in character and contains the climax, the binary form helps to create balance and organization in terms of both the production and resolution of tension in this fugue.


Despite the brevity of the coda, it operates as a summary of the fugue and also provides a final link to the inner form.  The texture is expanded to six voices, the outer two voices providing pedal points.  The upper two voices state the subject head in its inverted form while the lower voices state the original subject head.  This summarizes the fugue by offering the subject and its inversion in stretto, highlighting the important motivic and developmental material.  The thickness of texture and use of eighth note motion also provides a resolution to the tension created in mm. 33-38, the climax of the fugue.  In the G-major fugue WTC XV, analyzed by Tina Depko, the coda also functions as a resolution of the climax.  The major difference between these two codas, however, is that in this fugue the coda utilizes subject material whereas the G-major fugue does not.  Finally, the harmonic motion of I – V/V – I accompanied by a repetition of the fundamental line’s descent provides a last link to the inner form of this fugue.  This is illustrated in Figure 6.


Fig. 6



The binary form of this fugue is clearly articulated by the perfect authentic cadences in mm. 20-21 and mm. 42-43 and the repetition of the closing sections.  This is further reinforced by an emphasis on the original form of the subject in the first section and inverted form in the second section.  The points of imitation facilitate development and simultaneously saturate the fugue with the motivic material of the first four measures, providing a sense of unity.  There is little relief from the subject in this fugue and, as a result, the outer form serves to develop the fugue while the inner form provides the necessary balance and organization.




William Rothstein, Phrase Rhythm in Tonal Music, (New York: Schirmer, 1989), 104.

2[Back]Heinrich Schenker(1), Free Composition, trans. Ernst Oster (New York: Longman, 1935), 143

3[Back]Heinrich Schenker(2), Free Composition Supplement: Musical Examples, trans. Ernst Oster (New York: Longman, 1935), Fig. 156.

4[Back]Tim Smith, “Fugue in d minor,” Well Tempered Clavier by Johann Sebastian Bach: Twenty-seven Fugues and Select Preludes, 2002, (Oct. 9, 2003).

5[Back]As suggested to me by Dr. W. Renwick in an e-mail, Nov. 17, 2003.

6[Back]Schenker(1), 87.

7[Back]Schenker(2), Fig. 156.

Copyright 2003 by Jennifer Taylor