Definition
Two lines are in invertible counterpoint (or double counterpoint) when they can swap positions: the upper voice becomes the lower one and vice versa, and the result is still correct. In the most common case, at the octave, one of the two voices is moved by an octave (or by two: "at the fifteenth") so that it passes above or below the other. This is the technique that lets Bach reuse the same thematic material in every register of a fugue or invention without writing new material.
The inversion table
When you invert at the octave, every simple interval turns into its inversion: the two numbers always add up to 9.
| Original interval | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| After inversion | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
Practical consequences:
- Thirds and sixths swap with each other: they are the safest material, because they remain imperfect consonances.
- Octaves and unisons swap: they remain perfect consonances, but should be used sparingly (beginning, end, passing moments).
- The fifth becomes a fourth: this is the central problem. The fourth against the bass is dissonant in tonal style, so in double counterpoint at the octave the fifth must be treated as a dissonance: only as a passing tone, as a neighbor tone, or prepared and resolved.
- Second ↔ seventh: dissonances in both positions. Suspensions change nature: the 7–6 suspension (upper voice) becomes 2–3 (bass suspension), both correct. The 4–3 suspension, however, becomes 5–6, which is no longer a suspension: avoid it in lines meant to be inverted. The 9–8 suspension becomes 7–1 (a seventh resolving to the unison): weak, best avoided.
Working rules
- Keep the two voices within an octave of each other (or within a fifteenth if you invert by two octaves), otherwise the inversion does not really swap them.
- Avoid consonant fifths on strong beats and bare fifths at cadences.
- Watch out for hidden octaves and fifths: a fifth reached by similar motion becomes a fourth reached by similar motion, acceptable between upper voices but not against the bass.
- Always check both versions: write the combination A above B, then rewrite B above A and check every beat.
- Rhythmically, the two lines must be complementary (when one moves the other holds), so the inversion does not produce clashes.
Example in C major
Subject (upper voice, quarter notes): E–F–G–A–G–F–E–D | C
Countermelody (lower voice, half notes): C–D | E–B | C
When you build the countermelody, choose only thirds, sixths and octaves on strong beats (here E above C = third, G above E = third, C above C = octave) and use fifths only as passing tones. When you invert (countermelody an octave higher), E/C becomes C above E = 6: correct. If you had put G above C (a fifth) on a strong beat, after the inversion you would get C above G, a fourth against the bass: an error.
In real music
- Bach, Invention No. 1 in C major BWV 772: the motif and its counterpoint swap between right and left hand in the different sections.
- Bach, The Well-Tempered Clavier I, Fugue in C minor BWV 847: the subject and two countersubjects are written in invertible (triple) counterpoint at the octave, so they appear in different arrangements throughout the piece.
- Mozart, finale of Symphony No. 41 "Jupiter" K. 551: in the coda, five themes combine in multiple invertible counterpoint.
- Pop and soundtracks: when an arranger has the strings' countermelody played by the voice at some point and the melody played by the strings (a classic in final choruses), they are using the same principle: the two lines must work in both arrangements.
Exercise for a producer
- In FL Studio, in the Piano roll, write a 2-bar melody in C major (channel 1) and a slower countermelody (channel 2), using only thirds, sixths and octaves on strong beats.
- Duplicate the pattern and transpose the countermelody by +12 semitones (Ctrl+up arrow): it now sits above the melody.
- Listen to both versions and look for exposed fourths against the bass or empty fifths at cadences: fix them.
- In the track, use version A in the verse and version B in the final chorus: same material, different color.
Related topics
Counterpoint (first to fifth species), Canon and imitation, Invertible counterpoint at the tenth and twelfth, Fugue: countersubject and episodes, Two-part invention.