Invertible counterpoint (also called double counterpoint) is written so that either voice can be placed above the other without creating forbidden parallels. This technique is the engine of fugal episodes and developmental sections.
What Invertible Counterpoint Is
Invertible counterpoint is a two-voice passage in which the voices can be transposed relative to each other (one goes up, the other goes down, or one is moved an octave) and the result remains grammatically correct — no new parallel fifths or octaves arise, and no new dissonance treatments violate the rules. When the inversion is accomplished by moving one voice up (or down) an octave, it is called invertible counterpoint at the octave (the most common type). When one voice is moved by a tenth, it is invertible at the tenth; by a twelfth, at the twelfth.
The reason invertible counterpoint is valuable: in a fugue, the same two-voice passage can appear multiple times with the voices reversed, providing harmonic and textural variety without requiring new material. Bach's ability to derive seemingly infinite variety from a small amount of material rests largely on his mastery of invertible counterpoint. The two-voice passages in the episodes of his fugues are typically invertible — they appear in one order, then inverted, then at a new pitch level, then inverted again at yet another level, creating a sense of constant freshness from a small motivic kernel.
The Mathematics of Interval Inversion
The mechanism of invertible counterpoint depends on the predictable behaviour of intervals under inversion. When two voices are separated by some interval and one voice is moved an octave, the interval changes according to a fixed rule: at the octave, intervals invert by subtracting from 9.
- Unison (1) inverts to octave (8) — perfect consonance remains perfect consonance.
- Second (2) inverts to seventh (7) — dissonance remains dissonance.
- Third (3) inverts to sixth (6) — imperfect consonance remains imperfect consonance.
- Fourth (4) inverts to fifth (5) — dissonance (in bass) inverts to perfect consonance (or vice versa).
- Fifth (5) inverts to fourth (4) — perfect consonance inverts to dissonance (in bass).
- Sixth (6) inverts to third (3) — imperfect consonance remains imperfect consonance.
- Seventh (7) inverts to second (2) — dissonance remains dissonance.
- Octave (8) inverts to unison (1) — perfect consonance remains perfect consonance.
The critical problem: a perfect fifth (interval 5) inverts to a perfect fourth (interval 4). In two-voice counterpoint, a fourth between the bass and an upper voice is dissonant. Therefore, if the original counterpoint contains a perfect fifth with voice 1 above voice 2, the inversion will create a perfect fourth with voice 2 above voice 1 — which may be problematic as a bass interval. The solution: write the original counterpoint so that perfect fifths appear only where they can become acceptable fourths in inversion (typically as passing tones or as the upper note of a suspension chain).
Invertible Counterpoint at the Tenth and Twelfth
Invertible counterpoint at the tenth uses different inversion arithmetic: intervals are subtracted from 11. A third inverts to an octave, a fifth inverts to a sixth, a sixth inverts to a fifth, a seventh inverts to a fourth. The result: parallel thirds in the original become parallel octaves in the inversion, which is prohibited. Therefore, invertible counterpoint at the tenth cannot use parallel thirds (or parallel sixths, which produce parallel octaves under inversion at the tenth). This severely constrains the original writing — typically the voices must move predominantly in contrary or oblique motion.
Invertible counterpoint at the twelfth uses arithmetic of subtraction from 13. A fifth inverts to an octave (so parallel fifths become parallel octaves — also prohibited), a third inverts to a tenth, a seventh inverts to a sixth, and a sixth inverts to a dissonant seventh, so sixths must be treated as dissonances. The constraints differ from invertible counterpoint at the octave and tenth, and the student must master each set of rules separately. Bach uses all three types — invertible at the octave is most common, but the Goldberg Variations include canons at the second, third, fourth, fifth, sixth, seventh, eighth, and ninth, with corresponding invertible counterpoint principles.
Practical Procedure for Invertible at the Octave
To verify that a two-voice passage is invertible at the octave, follow these steps:
- List every interval between the two voices, beat by beat.
- Subtract each from 9 to get the inverted interval.
- Check whether the inverted intervals are all consonant or valid suspensions.
- Pay special attention to any fifths: in the inversion, they become fourths above the new bass. Are those fourths acceptable (passing-tone fourths, suspension-resolution fourths, or weak-beat fourths) or are they structurally dissonant fourths that need to resolve?
- Check that no parallel motion that was acceptable in the original becomes prohibited in the inversion. Parallel thirds become parallel sixths (fine in both). Parallel sixths become parallel thirds (fine in both). Parallel fifths become parallel fourths (both prohibited).
- If problems arise, revise the original passage to avoid the problematic intervals, or to place them in positions where their inversions will be acceptable.
Invertible Counterpoint in Fugal Episodes
The reason this technique matters so deeply is its role in fugal architecture. A fugal episode is typically a two-voice or three-voice passage that connects one subject entry to the next, modulating harmonically while developing motivic material. By writing the episode in invertible counterpoint, the composer can use the same material twice — first in one voicing, then with the voices swapped — without the listener perceiving mere repetition. The result is a passage that feels new but is in fact a transformation of material already heard.
The C minor fugue from Bach's Well-Tempered Clavier Book I (BWV 847) opens with a subject and countersubject in invertible counterpoint. When the subject returns in the bass, the countersubject appears in an upper voice; when the subject is in the upper voice, the countersubject is below. The same two-voice material works in both arrangements because Bach wrote it as invertible counterpoint from the start.
Sources and Further Reading
- Fux/Mann. The Study of Counterpoint. Norton. Chapter on Double Counterpoint.
- Gauldin. A Practical Approach to 18th-Century Counterpoint. Waveland Press. Chapter 9.
- Kennan. Counterpoint. Prentice-Hall. Chapter on Invertible Counterpoint.
- Aldwell and Schachter. Harmony and Voice Leading. Schirmer/Cengage.
- Open Music Theory — Invertible Counterpoint (openmusictheory.net, CC BY-SA 4.0).