Fugue by inversion, augmentation and diminution, counter-fugue and mirror fugue

Devices applied to the subject

Besides transposition, the subject can be transformed:

• Inverted (contrary motion): the intervals change direction. In C: C D E F becomes C B A G.
• Augmentation: doubled (or tripled) values.
• Diminution: halved values.
• Retrograde: read backward; rare in the fugue, frequent in canons.
• Combinations: inverted and augmented together, direct against inverted in stretto.

Each device may appear in a single entry or become the principle of an entire fugue.

Counter-fugue

In the counter-fugue (contrary fugue; Gegenfuge is the term used by Mattheson in 1739) the answer is the inverted subject, and the inverted form remains a protagonist throughout the fugue. Examples: The Art of Fugue, Contrapunctus 5, 6 and 7.

• Contrapunctus 5: many strettos between the direct and inverted forms.
• Contrapunctus 6, in French style: adds the two forms in diminution, with the dotted rhythm typical of the French overture.
• Contrapunctus 7: uses the subject in augmentation and in diminution, direct and inverted, simultaneously; the voice in augmentation runs through the fugue like a cantus firmus.

Fugue by augmentation

One voice presents the subject in long values while the others present it in normal or diminished values. The effect is that of a chorale over a contrapuntal animation. Technically it is a stretto in which the voices travel at different speeds: it must be checked like a mensuration canon.

Mirror fugue

Two fugues, one the mirror image of the other: every interval is inverted and the arrangement of the voices is also turned upside down (the soprano becomes the bass). Contrapunctus 12 of The Art of Fugue (in four voices) exists in a rectus form and an inversus form, and the mirror is complete. Contrapunctus 13 (in three voices) does the same, but the arrangement of the inverted voices is not exactly a mirror. Bach also left a version of it for two keyboards.

How to write an inversion that works

1. Choose an axis: the note around which to invert. In D minor the most convenient axis is often D (or the exchange D ↔ A).
2. Invert the subject and correct the accidentals: the leading tone C#, inverted around D, becomes Eb; often a diatonic form is chosen and altered afterward.
3. Check that the inversion makes tonal sense: it must be able to serve as an answer or as a subject in a closely related key.
4. Try strettos between the direct and inverted forms: they often work better than direct-direct strettos.

Example in D minor

Subject of The Art of Fugue: D A F D C# D E F. Inverted starting from A, with each interval flipped: A D F A Bb A G F (the ascending fifth becomes a descending fifth, the semitone below D becomes the semitone above A). This is the principle of the inverted form that Bach uses, with rhythmic and chromatic variants, in Contrapuncti 3 and 4 and in the counter-fugues.

Typical mistakes

• An inversion with accidentals that break the key.
• Augmentation that creates parallel fifths with the fast voices: the long subject must be treated like a cantus firmus, with the rules of the species.
• A mirror fugue checked only in the rectus form: every suspension, inverted, rises instead of falling.

Modern application

• Film scores: augmentation of a theme in the horns or trombones, under the same theme in normal values in the strings, is a classic way to give solemnity to the finale without writing new material.
• Production: the DAW functions for time-stretch (augmentation), inversion and retrograde apply these devices to a MIDI riff; use them knowing that, after the transformation, the harmony must be rechecked.
• Choirs: a subject in augmentation given to the basses on a long syllable supports the other sections singing the same theme in normal values.

Related topics

Counterpoint by contrary motion and mirror counterpoint, stretto, The Art of Fugue, types of canon, negative harmony.