Definition
In traditional invertible counterpoint the voices exchange register (top becomes bottom). In counterpoint by contrary motion each voice is turned upside down (melodic inversion: what rose now falls by the same number of scale steps) and the result must remain correct. When the entire composition is inverted, voices and registers together, we speak of mirror writing: this is the technique of Contrapunctus 12 and 13 of The Art of Fugue.
The trick of the axis
Inversion is done around an axis, a note that stays fixed. In the D Dorian scale (white keys) the ideal axis is D: the collection D E F G A B C is symmetrical around D, so the inversion turns white keys into white keys:
| Note | D | E | F | G | A | B | C |
|---|---|---|---|---|---|---|---|
| Inverted around D | D | C | B | A | G | F | E |
With this axis the vertical intervals keep the same number (a third stays a third, a fifth stays a fifth), so a correct counterpoint stays correct. What changes is the position of the semitones, and therefore the modal character.
Example in D Dorian
First species, counterpoint above the cantus:
- Cantus: D4 F4 E4 D4 G4 F4 A4 G4 F4 E4 D4
- Counterpoint: A4 D5 C5 D5 B4 A4 C5 E5 D5 C#5 D5
Inverted around D4 (the voices also exchange position):
- Inverted cantus: D4 B3 C4 D4 A3 B3 G3 A3 B3 C4 D4
- Inverted counterpoint: G3 D3 E3 D3 F3 G3 E3 C3 D3 Eb3 D3
Intervals: 5, 6, 6, 8, 3, 3, 3, 6, 6, 6, 8, identical to the original. The leading tone C#, inverted, becomes Eb: the cadence Eb–D under C–D is a Phrygian-type cadence. The mirror turns a cadence with a leading tone into one with a descending semitone.
Practical rules
- Choose the axis so that the collection of notes transforms into itself (D for the white keys; in harmonic minor adjustments are needed).
- Avoid intervals that change quality in a harmful way under inversion: the diminished fifth B–F, inverted around D, becomes the augmented fourth F–B.
- Prepared dissonances (suspensions) change direction: a suspension that resolves downward, turned upside down, resolves upward. In mirror writing, therefore, you use suspensions that are correct in both versions, or you limit the suspensions.
- Check the accidentals: C# ↔ Eb, F# ↔ Bb (around D).
In the repertoire
- The Art of Fugue, Contrapunctus 12 (in four voices) and 13 (in three voices): each fugue exists in rectus and inversus form. In 12 the mirror is complete (the soprano becomes the bass); in 13 the arrangement of the inverted voices is not mirrored exactly.
- Fugues by contrary motion (counter-fugue): the answer is the inverted subject, as in Contrapunctus 5–7.
- Canons by contrary motion in The Musical Offering and the Goldberg Variations (variations 12 and 15).
Typical errors
- Using an axis that produces continuous accidentals: the inverted version becomes chromatic and loses the mode.
- Forgetting that the cadence changes nature (from authentic to Phrygian).
- Thinking that inversion and retrograde are the same thing.
Modern application
- Film composition: a theme and its mirror are a natural pair for two characters in conflict; the D axis on the white keys guarantees that both stay in the same mode.
- Electronic production: the inversion (flip) functions of piano rolls with the axis on the tonic carry out exactly this technique; choosing the axis on D in C major keeps the inverted melody diatonic.
- Negative harmony: the mirror with a fixed axis is a close relative of negative harmony, which inverts chords around an axis between tonic and dominant.
Related topics
Invertible counterpoint at the octave, fugue by contrary motion and mirror fugue, negative harmony, The Art of Fugue, types of canon.