Balancing Rectus and Inversus in J.S. Bach’s Fugue XV from WTC I

Tina Depko


The definition of a fugue is anything but concrete. The fugues analyzed in this edition of the colloquium alone largely vary from one another, proving that writing a stable definition is in fact, impossible.


Perhaps a better way to look at the meaning of fugue is to trace the term back to its linguistic roots. The root of the word fugue in Latin is “fuga,” which means “flight” or “escape.” We can think of this as the opening voice taking off, leaving the other voices to follow.  The word “fugue” can also be traced to the German verb, “fugen” meaning “to assemble” or “to put together meticulously.”

Fugue XV in G major from Book I of the Well Tempered Clavier by Johann Sebastian Bach can aptly be described by the latter linguistic link.  Bach took great care when composing this fugue, keeping an overall formal idea in mind.

Overall Formal Idea

Rectus and inversus is the overall formal idea in this fugue. The term rectus means to present a subject in its original form, while inversus means to present a subject upside down. The Art of Fugue is representative of Bach’s fascination with formal possibilities and Fugue XV is, “designed formally around the possibilities for development in rectus and inversus forms."1


An emphasis on balance between the amount of rectus and inversus material is evident in Fugue XV. The exposition contains three presentations of rectus subject material in G major, and the counter exposition creates balance with three inversus subject entries in the same key. The development has two subject entries in E minor, including one that is inversus and one that is rectus. The following four entries in the development, which include a pair of rectus subjects in d minor and another pair of rectus subjects in D major, create a higher number of rectus entries. Balance is restored in the closing section with an inversus entry in D major, and the final entries of two inversus subjects and one rectus subject. The total number of rectus entries is nine and the total number of inversus entries is seven. Although these numbers are not equal, the placement of an inversus counter exposition after a rectus exposition, and a closing section with three inversus entries to balance the five rectus entries in the development indicate an attempt to create balance. See Example 1.0.



Example 1.0

Balance Between Rectus and Inversus


Rectus Inversus
Exposition (3)
Counterexposition (3)
(G major, D major, G major)
(G major, D major, G major)
Development (5)
Develepment (1)
E minor
E minor
B minor
Closing section (3)
B minor
D major
D major
G major
D major
G major
Closing section (1)

G major

Total rectus (9)
Total inversus (7)

The Basics


This piece is in G major and is a three-voice fugue. I label the top voice soprano, designate the middle voice as alto and label the low voice the bass.  These labels are in accordance with the position of the soprano and alto in the treble clef. The lowest voice is designated the bass because of its substantial use of low notes in the bass clef.


Fugue XV is in three-part form, as there is an exposition, a development and a closing section.  While there is a counter exposition, it is an extension of the exposition because the key remains in G major and the subject material is inversus rather than developed.


The distribution of rectus and inversus material in the sections can be seen in

Example 2.0


Exposition (mm. 1 to 19)

The exposition of Fugue XV encompasses mm. 1 to 19. The subject enters in the soprano in G major in m. 1. The first three measures of the subject ascend, while the fourth measure descends. The subject features a characteristic rhythmic pattern in m. 1 and m. 4 and a contrasting rhythmic pattern in m. 2 and m. 3. The subject uses the notes of the G major triad, allowing the melody to be inverted.



Figure 1.0

Subject of Fugue XV



Where the subject ends is an issue of contention among scholars like Siglind Bruhn, but after careful consideration it would appear the subject ends on the F# on the first beat of m. 5.


An argument could be made for designating the end of the subject on the final G in m. 4. However, designating the subject end on a weak beat does not make sense rhythmically, as the listener is waiting for resolution on a strong beat.


The answer enters in m. 5 in the alto in D major and the final entry of the exposition occurs in m. 11 in the tonic in the bass.  All three entries in the exposition are rectus.


The accompanying music in mm. 5 to 8 is rhythmically opposed to the answer, as its sixteenth notes juxtapose the eighth notes of the answer. Bach reuses it to accompany other subject entries in a free manner when convenient, similar to a countersubject.


Bach does not use a cadence at the end of the exposition as the counter exposition begins in the same key after episode 1.


Bridge and Episodes


In order to understand how Bach connects various rectus and inversus presentations and provides relief, we need to examine the bridge and the episodes.  Fugue XV contains a bridge (mm. 9 to 10) and seven episodes.


Figure 2.0




Measures 9 and 10 serve as a bridge, linking the answer in D major and the final subject entry in G major in the exposition. Besides changing the key from D major back to G major, the bridge also contains a melodic idea that reappears in each of the seven episodes and the coda.


This melodic idea is introduced in m. 9 in the soprano, descending. I refer to this melodic idea as the leaping motive. The leaping motive is placed in the alto in m. 10, where it ascends. It reappears throughout the piece in either ascending or descending form.  I label each episode, therefore, as A.


The leaping motive is one of three motives that create triple counterpoint in four episodes. I label the leaping motive as A, the pair of sixteenth notes with the quarter note and descending sixteenth note pattern as X, and the eighth note and dotted note sequence as Y.


The use of A, X and Y motives in triple counterpoint allows us to understand the episodes as Episode 1 (A+X+Y), Episode 2 (A+X+Y), Episode 4 (A+X+Y), Episode  6 (A+X+Y). See Example 3.0. The excluded episodes, including Episodes 3, 5 and 7 will be discussed in the following paragraphs.


 Example 3.0

Episodes with Triple Counterpoint


Episode 1
Episode 2
Episode 4
Episode 6

            I label the bass material in episode 1 as “A” because it is derived from the bridge in mm. 9 and 10. Themes “X” and “Y” are new material. See Figure 3.0.

Figure 3.0

Triple counterpoint in Episode 1, Episode 2 and Episode 4



The leaping motive A is also used above or below an ascending or descending sixteenth note sequence that I label as Z in several of the episodes. We can understand this combination as Episode 2 (A+X+Y+Z), Episode 5 (A+Z), Episode 7 (A+Z).


Episode 3 is similar to mm. 9 and 10, featuring the leaping motive, but no triple counterpoint with X or Y sequences, or ascending/descending sixteenth note Z sequences. Therefore, I simply label this episode as A.


The carefully planned episodes are an important feature of this fugue.  The subject material of Fugue XV is carefully planned around rectus and inversus, while the episodes are planned around the leaping motive. The result is formal rectus and inversus sections alternating with unified episodes. This could suggest a loose link to rondo form. The comparison of this fugue to rondo form may be extreme for some analysts, as there are several problems, such as the fact that the subject material is not restated exactly at each entry. However, it is a worthwhile factor to consider and emphasizes the sense of balance in this formally designed piece.


Counter exposition (mm. 20 to 37)


See Example 2.0


The rectus exposition is balanced by the inversus counter exposition2 at mm. 20 to 37. There are three rectus entries in the exposition in G major and three inversus entries in the counter exposition in the same key.


The first inversus subject enters at m. 20 in the alto in the tonic key and the inversus answer starts at m. 24 in the soprano in the dominant key. The final inversus subject enters at m. 28 in the bass in the tonic. The entrance of all three voices in the counter exposition makes it a complete counter exposition. Following the exposition and counter exposition, there is now an exact balance between the three rectus and three inversus entries.


Development (mm. 38 to 69)


See Example 2.0


The development section features three pairs of subject entries, including five rectus and only one inversus. The large number of inversus entries in the closing section provides balance to the rectus-dominating development.


The development begins at m. 38 when the first subject enters rectus in m. 38 in E minor in the soprano. Balance is created by the entrance of an inversus subject at m. 43 in the alto in the same key.


A second set of entries starts at m. 51 with a rectus subject in the soprano in B minor.  Another B minor rectus subject entry commences in m. 52 in the bass before the previous subject is complete, resulting in stretto. The use of the soprano and bass without the alto results in partial stretto.  


The final set of entries in the development commences at m. 60 with a rectus subject entry in the alto in the key of D major. The second statement enters in m. 61 in the soprano with variations before the first previous subject is complete, creating partial stretto. A pedal point in the bass holds the dominant note of A in mm. 62 and 63. This pedal point creates a fuller texture but still allows the partial stretto to be heard.  See Figure 4.0.


Figure 4.0

Stretto in D major



The development ends at the perfect authentic cadence in m. 69. It serves to mark the beginning of a new section, as “cadence is frequently a principal means of defining the sections of a fugue, since cadence is the primary means of indicating the close of a given section of the work.”3 The cadence is an elision as it marks the end of the development and the beginning of the closing section with the entrance of the subject in m. 69 in the bass.  See Figure 5.0.


Figure 5.0

Cadence at m. 69



Closing section (mm. 69 to 86)


See Example 2.0


The closing section balances the development’s excessive rectus entries with an emphasis on inversus subject entries. The first subject entry of the closing section is inversus in the bass in m. 69. The key remains in D major


The final group of subject entries starting at m. 77 is a fusion of inversus and rectus in the key of G major.  The first subject enters in the alto and is inversus. Subject variations enter in m. 78 in the bass, also inversus. The final subject enters in m. 79 in the soprano. Unlike the previous two entries, this final subject is rectus. The close proximity of the three voices results in complete stretto. An emphasis on rectus and inversus occurs in m. 77 with homophony. This homophony consists of a rectus subject in the soprano, an inversus subject in the alto and an inversus subject in the bass. The use of homophony in this measure builds tension, driving to the climax and ultimately the piece’s conclusion.


The climax of Fugue XV occurs at m. 81. The wide range of this measure signifies a climax with a Bb in the upper register of the soprano and a low E in the bass. Besides the range, the presence of chromatic notes in both the bass and soprano also indicate a climatic point. This chromaticism creates dissonance and there is also motivic complexity as the subject in the soprano and alto comes together with the leaping motive in the bass for the first time in the piece in mm. 80 and 81. See Figure 6.0.



Figure 6.0

Climax at m. 81



Although there is a climax in this piece, there is not a single wave of tension. Instead, Fugue XV has waves of tension. If we think of tension as an arch, with the low points representing release and the higher points representing tension, the waves can be shown in this piece by looking at each section.
"Bach's fugues are all constructed with very carefully-balanced ‘sections.’  Since this design or structural layout is slightly different in each fugue and of vital importance to an understanding of the composition, it is useful to develop some secure method of analysis."4


Each section has its own wave of tension, which the exception of the development, which has two waves of tension.


In the exposition, the tension curve starts at m. 1 and slowly escalates as the texture thickens from one voice to two voices with the entrance of the answer and its accompaniment. The curve continues upwards with the presence of all three voices in close proximity in the final subject entry of the exposition, peaking at m. 15. The curve descends as tension is released in episode 1 and slowly returns to the starting point at m. 19. Episode 1 releases tension through triple counterpoint, with two main themes, A and X far apart in range, and with the limited use of Y.


The tension curve starts climbing again at the beginning of the counter exposition and ascends with the three-voiced texture until midway through the inversus answer at m. 25 when the three voices are close together. The remaining bars of the counter exposition feature two voices with a larger range, creating a release of tension that continues through to the end of episode 2. Triple counterpoint in episode 2 relieves tension, as it did in episode 1.


The first tension curve of the development starts at m. 38. The curve ascends very little as episode 4 relieves any tension that was created by the two subject statements in e minor with triple counterpoint, but the stretto at m. 51 creates great tension and causes the tension curve to ascend. Episode 5 relieves the tension from the stretto with a thin, two-voiced texture with a large range and a pedal point at mm. 56 and 57, and brings the curve back down. The second curve of the development starts at m. 60 with tension from stretto in D major. This tension is released a few bars later is episode 6 with triple counterpoint and the curve ends at the cadence at m. 69.


The closing section features a single curve that stays relatively stable until complete stretto and homophony in a narrow range in m. 79 causes the curve to ascend sharply until it peaks at the climax at m. 81. The curve descends in the remaining bars as the coda uses the leaping motive from the tension-relieving episodes and a wide range.


Coda (mm. 83 to 86)


A perfect authentic cadence at mm. 82 and 83 in G major marks the beginning of the coda. The purpose of this coda is to release the tension that peaks during the climax in m. 81. The coda does not use any subject matter, and therefore, it does not have a link to rectus and inversus. However, the coda is linked to the piece by the leaping motive that is first introduced in m. 9 and 10. This leaping motive is apparent from m. 83 in the soprano where it is descends, before it transfers to the bass in the same measure. The motive returns to the soprano in m. 84 and then appears in the newly introduced tenor. The leaping motive is placed back in the soprano in m. 85 where it continues to ascend before the closing at m. 86.  The placement of the leaping motive creates balance among the voices in the coda. Linking the coda to the possible connection to rondo, the leaping motive is used in mm. 83 to 86 because the subject material was just stated at the end of the closing section. Also, the use of the leaping motive in episodes to generally relieve tension makes its presence in the tension-relieving coda unsurprising. The return of the leaping motive in the tonic key is a satisfactory ending.


A pedal point appears in the bass at m. 83. This pedal point is on the A and is held to the end of the piece. Another tonic pedal point enters an octave higher in the newly added tenor in m. 85, reinforcing the pedal point. A pedal point on the tonic note of G in the key of G major is not surprising, as “the tonic pedal point most typically signals the end of a fugue.”5







The coda in this piece is different than the coda in Fugue VI in D minor, analyzed by Jen Taylor. The coda in fugue VI is two measures in length, rather than four measures and uses both the subject and inversion, unlike Fugue XV where there is no rectus and inversus subject material. The coda in Fugue VI is also different as it uses six voices, including two pedals in the outer voices and four inner voices.


Fugue XV is a balance of rectus and inversus. The use of the technique in each of the four sections of the piece, including the exposition, counter exposition, development and closing section, emphasizes the overall formal idea. A balance between the number of rectus and inversus entries is created through an almost equal number of each. The leaping motive unifies the episodes, creating a fugue that has formally designed subject entries and episodes.



1[Back]Renwick, William, Encyclopedia Entries on Fugue, (Unpublished Manuscript, 2003), Rectus and inversus, 80.

2[Back]Smith, Tim and David Korevaar. The Well-Tempered Clavier, ADDRESS: (October 18, 2003).

3[Back]Renwick, Cadence, 10.

4[Back]Bruhn, Siglind. J.S. Bach’s Well-Tempered Clavier: In Depth Analysis and Interpretation, Vol. III, (Hong Kong: Mainter International, Ltd., 1993), 53.

5[Back]Renwick, Pedal Point, 67.

Copyright 2003 by Tina Depko