The Fugue

Based on Hugo Norden's Foundation Studies on Fugue


About the 'Art of Fugue' you may also read:

Glenn Gould Discourses on Fugue: Watch and Learn by John Stone

 

Search for sheet music on the Art of Fugue

The Bach's Art of Fugue in MIDI files


I
The Subject

A fugue consists of the statement and development of a single subject (the main idea). It usually is stated at the beginning of the fugue by a single voice.

The subject usually has following three characteristics:
- it is usually brief, but represents a complete musical idea;
- it is easy recognizable in contrapuntal texture;
- it is easy to manipulate (harmonic, melodic, and contrapuntal) and to exploit.

The opening of a subject ('head' of the subject or 'La tęte du sujet') should be strong and easy to recognize.

 

II
The Answer

In the simplest case, the answer is the subject (dux) transposed to the dominant key (5th higher or 4th lower).
If a subject in the major key contains a leading tone, the answer (comes) requires an accidental for its leading tone.
If a subject is in the minor key, the answer requires accidentals for both the II and VII notes of the dominant key.
If the subject contains additional accidentals, these too must be reflected in the answer.
An answer that is a literal transposition (every note is transposed exactly a perfect 5th up or a perfect 4th down) of the subject to the dominant key is identified as a REAL answer.
When notes in the subject are in the dominant key and are answered in the tonic key in contrast to the rest of the subject, the resulting answer is said to be TONAL.

 

Subjects That Require a Real Answer
Those subjects will begin and end on the I or III notes, and they will not contain a leap from the I or III notes to the V or VII notes at or near the beginning.

Subjects That Require a Tonal Answer

Group I
- Subjects of this kind will begin with one or more notes in the dominant key and complete the subject in the tonic key. The parts that are in the dominant key will be answered in the tonic key, and vice versa.

Group II
- Subjects of this kind start in the tonic key and end in the dominant key, while their answers do exactly the opposite.

Group III
- Subjects of this kind start in the dominant key, have a middle section in the tonic key, and end in the dominant key.

Group IV
- The subject contains a leap from the I or II to the V or VII notes at or near the beginning, or • The entire subject is answered in its subdominant key. Subjects answered this way usually begin on V or VII notes.

Modulating Effects to the Dominant Key
In order for a subject to end on V or VII, a modulation from the tonic key (either the opening or the middle section of the subject) has to take place. Following thematic devices serve this purpose:
- The modulation takes place at an ascending major 2nd from accented to unaccented notes
- It takes place between repeated notes from accented to unaccented notes
- In a conspicuous leap, usually (but not necessarily) from unaccented to accented notes
- After a tied or dotted note or rest, provided it does not damage a suspension
- In a descending minor 2nd, often from accented to unaccented notes

 

III
The Countersubject

Countersubject means countermelody, counterpoint (punctus contra punctum). In a fugue it is the counterpoint to the subject appearing simultaneously with the comes.
The countersubject must be so constructed that it will function equally well both above and below the subject and the answer (invertible counterpoint).

Process for Composing a Countersubject

1) All of the subject is in the tonic key:
Those subjects require a real answer (literal transposition). But, these in turn subdivide into two classes of contrapuntal construction:
- those without tied notes, and
- those with tied notes

Subjects without tied notes:
- the longer note, whether in the subject or in the countersubject, will serve as Cantus Firmus; the part with faster notes will act as counterpoint
- when the countersubject is transposed to its dominant key, it automatically provides the countersubject for the answer

Subjects with tied notes:
- the suspensions and retardations in the subject should be treated first
- then all the suspension opportunities against the subject should be treated
- finally, the remainder of the countersubject should be composed and the suspensions embellished if desired

2) Most of the subject is in the tonic key:
Set up a three-stave Double Counterpoint format with the subject on the top line, and on the bottom line the answer transposed to its subdominant key. Fill in the countersubject on the middle line so that it will produce correct counterpoint against both subject and answer. Copy out the countersubject in Double Counterpoint against both subject and answer (transposed back to its original key). The countersubject, too, must be transposed to match the key of the answer. Although the countersubject is the same melodically for both subject and answer, its contrapuntal operation is different. If the subject begins on a dominant note, the countersubject should begin before the subject in order to make a suitable approach to the limited contrapuntual resources that are available in Double Counterpoint at the 9th (The answer transposed to its subdominant key and the subject form a 9th in Double Counterpoint format, in between which the countersubject must be added).

3) Most of the subject is in the dominant key:
Set up a three-stave Double Counterpoint format with the subject on the bottom line, and the answer transposed to its dominant on the top-line. The dominant portion of the subject thus will relate in Double Counterpoint at the 8th ve while the notes in the tonic key relate at the 9th. Fill in the countersubject on the middle line so that it will provide suitable counterpoint against both subject and answer. Copy out the countersubject in Double Counterpoint against both subject and answer (both countersubject and answer transposed to its original key).

 

IV
The Stretto

Stretto is the contrapuntal device wherein the subject or answer is played against itself canonically, or when the subject and answer are made to overlap. It is most necessary in a stretto formation that all entries of the subject or answer be stated with all intervals being used literally. The themes can be modified intervallically or used only partially.

1) Literal Stretto
The subject is treated canonically, the answer does not appear. It can be structured as a canon at the 8ve, in Double Counterpoint at the 15th. This process will automatically produce one canon at the 8ve above, and another canon at the 8ve below.

2) Stretto by contrary motion
This kind of a stretto can be constructed by exact intervallic imitation in contrary motion (an ascending minor 2nd can just be imitated by a descending minor 2nd) or by inexact imitation (an ascending minor 2nd can be imitated either by a descending minor 2nd or a descending major 2nd).

3) Retrograde Stretto
The subject or answer is played backwards against itself (Crab Canon or Cancrizans Motion). Three reasons why this kind of stretto is impractical:
- it cannot be recognized audibly,
- many rhythms become disjointed and ineffective when heard backwards, and
- it is likely to set up awkward harmonic situations.

4) Stretto by Augmentation: the subject is played against the subject in augmentation (the note values are doubled, tripled, etc.).

5) Stretto by Diminution.
The subject is played against the subject in diminution (smaller note values).

6) Stretto in Contrary Motion and Retrograde.
This kind of stretto entails a crab canon in contrary motion.

7) Stretto in Contrary Motion and Augmentation
This kind of stretto entails an augmentation in contrary motion.

8) Stretto in Contrary Motion and Diminution.

9) Stretto in Retrograde and Augmentation.

10) Stretto in Retrograde and Diminution.

11) Stretto in Contrary Motion, Retrograde, and Augmentation.

12) Stretto in Contrary Motion, Retrograde, and Diminution.

 

Solving the Stretto Problem
Constructing Strettos to a given subject, as in a fugue examination, and constructing an original subject to yield pre-planned stretto combinations.

1) Constructing strettos to a given subject:
Fit the beginning of the subject against the end of the subject in as many ways as possible.

2) Composing a subject with pre-established stretto requirements:
Formulate the problem (for example compose a subject of type real in the key of D-major, 2/4 rhythm, 3 measures long, to yield a stretto at the 8ve above after 1 measure).

Set up the canon problem so that the first and last notes will conform to the terminal specifications stated before.
Complete the canon.

 

V
The Codetta and Episode

The terms codetta and episode both refer to connecting passages that are generally composed of free material that may be either similar to or different from motives in the subject or countersubject. The codetta is used to lead to a new entrance of the subject or answer in the exposition, while the episode is designed to link entries of the subject or answer after the exposition is completed.

Codetta
The first codetta appears at the end of the initial statement of the subject when there is a time lag between the end of the subject and the beginning of the answer. The codetta between the second and third entries is usually longer and more substantive than the one between the first and second entries. In a four-voice fugue there can also be a codetta between the third and fourth entries.

Episode
The episode differs from the codetta chiefly in its location in the fugue, which is after the exposition (all voices have come in with the subject and answer for the first time).

Process for the Writing of an Episode
The harmonies at the two ends of the episode must be clearly defined (those two ends are the ending of one subject and the beginning of the other subject). Decide on a contrapuntal or harmonic device to direct the contrapuntal texture from the beginning to the end of the episode (for example a canon in contrary motion derived from the opening motive of the countersubject). Connect the episode artistically at both ends so that there are no 'seams' or 'rough edges'.

 

VI
The Pedal-Point and Cadenza

Pedal-Point
The purpose of a pedal-point is to reestablish the tonality of the composition after it has become clouded by the numerous modulations and digressions along the way within the middle entries of the subject and answer and in the connecting episodes.

Cadenza
The purpose of the cadenza is to halt the contrapuntal flow of the fugue, and to bring the composition to a satisfactory close. The cadenza is generally made up of new material that does not appear in the subject (other than the episode and the pedal-point)

 

VII
The Exposition

The opening section of the fugue wherein each voice is brought in singly, alternating between subject and answer.
Following is one example of a possible exposition in a four-part fugue:

S
subject
countersubject

A
answer
countersubject
free part

T
subject
countersubject
free part
free part

B
answer

 

The Process of Writing an Exposition
Write the correct answer to a subject and compose a suitable countersubject. Decide on an entrance arrangement, and write in the subject, answer, and countersubject where desired. Complete the exposition by supplying the necessary codetta and free parts.

 

VIII
Harmony

The fugal harmony must be considered on three levels:
- the key scheme of the composition as a whole
- the underlying chordal structure
- the vertical relationship between the several voices

In order to see the underlying chordal structure, it is necessary to remove all embellishing materials.


All the material is based on Hugo Norden's Foundation Studies on Fugue.
Published on Kunst der Fuge by the consense of Christian Bohnenstengel.
The above texts or parts of them may not be published without prior permission of the Author.