A composer writes a three-voice texture in which any voice can serve as soprano, alto, or bass. How many distinct voice orderings must all produce acceptable counterpoint for the texture to qualify as triple invertible counterpoint?
A2 — the original ordering and one complete inversion
B3 — one for each voice serving as the bass line
C6 — all permutations of three distinct voices
D9 — each pair of voices can exchange positions independently
With three voices, the possible orderings (soprano/alto/bass assignments) are the 3! = 6 permutations of three objects. Genuine triple invertible counterpoint requires that ALL six permutations produce acceptable voice-leading. This is the combinatorial heart of the technique: each additional voice multiplies the number of required valid arrangements factorially, which is why the compositional constraints grow so rapidly.
Question 2 Multiple Choice
In counterpoint designed to invert at the tenth (a compound third), what does a perfect fifth (interval 5) transform to under voice exchange?
AA perfect fourth (interval 4) — as in standard inversion at the octave, using formula 9 − n
BA perfect fifth (interval 5) — perfect fifths are preserved under any inversion
CA sixth (interval 6) — using the formula 11 − n at the tenth
DAn octave (interval 8) — compound inversion doubles the interval value
At the octave the formula is 9 − n: a fifth (5) maps to a fourth (4). At the tenth, the formula changes to 11 − n: a fifth (5) maps to 11 − 5 = 6, a sixth. This different mapping means that intervals safe at the octave may be problematic at the tenth, and vice versa. The interval of inversion must be chosen first, then the transformation formula applied consistently to identify which intervals to use and avoid.
Question 3 True / False
Quadruple invertible counterpoint is more demanding than triple invertible counterpoint because it requires all 24 permutations of four voices to produce acceptable part-writing.
TTrue
FFalse
Answer: True
4! = 24 permutations must all satisfy contrapuntal constraints, versus 3! = 6 for triple invertible counterpoint. This factorial growth explains why writing genuine quadruple invertible counterpoint is among the most demanding compositional feats. Bach's Art of Fugue and Brahms's late chamber works contain celebrated examples; in practice, composers often exploit a subset of the 24 permutations rather than all of them.
Question 4 True / False
In invertible counterpoint at the octave, a perfect fifth between two voices transforms to another perfect fifth after the voices exchange positions.
TTrue
FFalse
Answer: False
At the octave, the transformation formula is 9 − n. A perfect fifth (interval 5) maps to 9 − 5 = 4 — a perfect fourth, not another fifth. This exchange between fifths and fourths is exactly why those intervals require careful handling: a fifth in the original (generally consonant in strict counterpoint) becomes a fourth (treated as a dissonance requiring resolution in many contrapuntal contexts) after voice exchange.
Question 5 Short Answer
Explain why the interval of inversion matters when composing invertible counterpoint, and how the transformation formula changes between inversion at the octave and at the tenth.
Think about your answer, then reveal below.
Model answer: The interval of inversion determines how each interval between voices transforms after the exchange. At the octave, when voices swap register, interval n becomes 9 − n: a third (3) becomes a sixth (6), a fifth (5) becomes a fourth (4). At the tenth, the formula changes to 11 − n: a third (3) becomes an octave (8), a fifth (5) becomes a sixth (6). Different mappings mean different intervals are 'safe' (consonant in both the original and inverted forms). A composer must choose the inversion interval first, then apply the appropriate formula to identify which intervals to use and avoid throughout the passage.
This is why analyzing invertible counterpoint requires identifying the intended inversion interval — the same passage may be technically correct for one inversion interval and full of violations for another. The formula change between octave and tenth arises from the different transposition the inversion produces on the interval stack.