Questions: Invertible Counterpoint and Multiple Counterpoint

Question 1

You are writing two voices intended for invertible counterpoint at the octave, and you use a perfect fifth between them. When the voices exchange registers, this interval becomes:

• A. A perfect fifth — intervals are preserved under octave inversion
• B. A perfect fourth — which is treated as dissonant against the bass in traditional counterpoint
• C. A minor sixth — because inversion reverses interval quality
• D. A tritone — because the perfect fifth and fourth differ by a tritone

Answer: (B)

Using the inversion rule for counterpoint at the octave — new interval = 9 − old — a perfect fifth (5) becomes 9 − 5 = 4, a perfect fourth. A fourth is treated as dissonant when it falls between the bass and an upper voice, creating voice-leading problems in the inverted arrangement. This is the central danger in invertible counterpoint: an interval that sounds fine in the original position can become problematic when the voices swap.

Question 2

Why do composers writing for invertible counterpoint at the octave favor intervals of thirds and sixths?

• A. Thirds and sixths are more consonant than other intervals in tonal harmony
• B. Thirds invert to sixths and sixths invert to thirds — both remain consonant in either arrangement
• C. Thirds and sixths eliminate parallel motion, reducing voice-leading errors
• D. Bach's strict style allows only thirds and sixths between inner voices

Answer: (B)

Using the 9-minus rule: a third (3) becomes 9 − 3 = 6, a sixth; a sixth (6) becomes 9 − 6 = 3, a third. Both thirds and sixths are consonant intervals, so they remain consonant after inversion. They are the 'safe' interval choices for invertible counterpoint — whatever arrangement the voices appear in, the intervals hold up. Fifths, by contrast, become fourths, creating dissonance against the bass.

Question 3

In invertible counterpoint at the octave, a major third in the original arrangement becomes a minor sixth when the voices exchange registers.

• T. True
• F. False

Answer: True

Yes. The interval number transforms by the rule 9 − 3 = 6, giving a sixth. When a major interval is inverted, its quality becomes minor (and vice versa): a major third inverts to a minor sixth, a minor third to a major sixth. This quality flip is part of what makes invertible counterpoint planning non-trivial — the composer must verify that the resulting quality in the inverted arrangement still makes harmonic sense.

Question 4

Any two voices that independently satisfy standard voice-leading rules can be used as invertible counterpoint, since each part is already correct on its own.

• T. True
• F. False

Answer: False

This is the key misconception. Two individually correct voices may produce voice-leading problems when inverted. The most dangerous case: a perfectly consonant fifth between the voices becomes a fourth — dissonant against the bass — when the voices swap. Invertible counterpoint requires planning both arrangements simultaneously from the start, not assembling two independently correct lines and hoping they invert cleanly.

Question 5

Explain why the '9 minus the original interval' transformation rule determines which intervals are safe to use in invertible counterpoint at the octave, and give an example of an interval that becomes problematic after inversion.

Model answer: When two voices exchange registers by an octave, every interval is replaced by its complement to 9 (counting from 1): a third becomes a sixth, a fifth becomes a fourth, etc. The composer must verify that the transformed interval is also acceptable voice-leading. The classic problem: a perfect fifth (consonant in any position) becomes a perfect fourth (9 − 5 = 4), which is dissonant when it falls between the bass and an upper voice. Parallel fifths are already forbidden; after inversion they become parallel fourths — equally problematic. Planning for inversion means choosing intervals whose complements to 9 are also consonant.

The 9-minus rule captures what happens geometrically when you flip the relative register of two voices: the interval that was above the bass is now above the soprano, and the sizes that fill an octave sum to 9 (e.g., a third and a sixth, a fourth and a fifth). This is why invertible counterpoint is considered advanced compositional skill: it requires thinking in two arrangements simultaneously — checking not just that the current layout works, but that its mirror image works too.