Double, triple and quadruple fugues

Definition

A double fugue has two subjects, a triple three, a quadruple four. The second subject differs from a countersubject in that it has an identity of its own and, often, an exposition of its own. All the subjects must be written in invertible counterpoint with each other.

The two types

1. Subjects exposed together: the second subject enters from the beginning, simultaneously with or immediately after the first, and the pair behaves as a block. Examples: Kyrie of Mozart's Requiem (double fugue; the countersubject derives from the final chorus of Handel's Dettingen Anthem), the fugue of Bach's Passacaglia in C minor BWV 582.
2. Subjects exposed separately: each subject has its own exposition; afterward, the subjects are combined. Examples: Fugue in Eb major BWV 552 (St. Anne) for organ, a triple fugue; Fugue in F# minor BWV 883 from The Well-Tempered Clavier II, a triple fugue with a subject and two countersubjects that are gradually added together.

The Art of Fugue as a catalog

• Contrapunctus 8: triple fugue in three voices, with independent expositions.
• Contrapunctus 9: double fugue at the twelfth: the two subjects are in invertible counterpoint at the twelfth.
• Contrapunctus 10: double fugue at the tenth.
• Contrapunctus 11: triple fugue in four voices on the three subjects of Contrapunctus 8 inverted.
• Contrapunctus 14 (Fugue on 3 subjects): unfinished; the third subject begins with the motif B–A–C–H (Bb A C B). The fugue breaks off at bar 239; tradition considers that it was meant to become a quadruple fugue with the return of the main subject.

How it is built

1. Write the first subject and the second against it, in double counterpoint at the octave (or at the chosen interval).
2. Contrast the two subjects in rhythm and profile (one in long values, one lively; one by step, one by leaps).
3. If the subjects will have separate expositions, make sure each has an answer (real or tonal) and its own logic of entries.
4. In the combination section check all the arrangements: with two subjects, two; with three subjects, six.
5. Plan the fugue's culminating point for the moment when all the subjects sound together.

Distinguishing a second subject from a countersubject

• The countersubject is born against the answer and follows the subject; it has no exposition of its own.
• The second subject has an exposition of its own or, if it appears from the beginning, is treated with the same dignity (strettos, isolated entries).
• Borderline case: in the Fugue in C# minor BWV 849 three themes combine in invertible counterpoint, but only the first has a true exposition; for this reason analysis considers it a fugue with countersubjects rather than a triple fugue in the strict sense.

Permutation fugue

A Baroque variant: each voice enters with the subject and then continues with two or more themes in the same order; there are no episodes and the combinations follow one another. Example: the final chorus of the cantata Himmelskönig, sei willkommen BWV 182.

Typical mistakes

• A second subject that is too similar to the first: the combination is not perceived.
• A final combination not checked in all arrangements.
• Separate expositions that are too long: the fugue loses unity before the combination.

Modern application

• Film scores: two themes for different characters presented separately and then superimposed in the finale are a dramatic double fugue; the moment of superimposition becomes the narrative climax.
• Musicals: ensemble numbers in which several characters sing different melodies that finally overlap apply the same idea (quodlibet).
• Production: in the ending of a song, layering chorus hooks and the verse melody works only if the two lines were conceived in double counterpoint.

Related topics

Triple and quadruple counterpoint, invertible counterpoint at the tenth and twelfth, The Art of Fugue, countersubject, school fugue.