In Bach's Musical Offering, one canonic piece makes perfect musical sense when the score is turned upside-down and read from right to left. What type of canon is this?
ACanon by inversion — the follower mirrors the melodic intervals upside down
BCrab canon (cancrizans) — the follower performs the leader's melody backward
CProportional canon — the follower performs at half the original speed
DRound — the voices enter at the unison after a fixed time interval
A crab canon (cancrizans) is defined by the follower performing the dux melody in retrograde — backward. The result is a piece that is palindromic in some sense: the score works equally well in reverse. A canon by inversion flips the melodic intervals vertically (ascending becomes descending), which is a different transformation. A proportional (mensuration) canon varies tempo, not direction.
Question 2 Multiple Choice
A composer wants to write a mensuration canon where the soprano sings the melody at normal speed while the alto sings the same melody at half speed. Which skill from invertible counterpoint is most critical, and why?
AThe ability to write florid ornamentation, since the slower voice needs more notes to fill time
BThe ability to compose a melody that produces legal vertical intervals with itself at every possible temporal offset, since voices will be at different positions in the melody simultaneously
CMastery of voice-leading in parallel motion, since both voices share the same pitches
DKnowledge of modal harmony, since proportional canons require a different harmonic language than tonal music
In a mensuration canon, while the soprano sings beat 10 of the melody, the alto (at half speed) is singing beat 5. These two melody points sound simultaneously, so the vertical interval between them must be acceptable counterpoint. This must hold at EVERY temporal offset throughout the piece — meaning the melody must produce legal counterpoint with every displaced version of itself. This is precisely invertible counterpoint applied temporally: the constraint that any point in the melody can serve as either the upper or lower voice against any other point.
Question 3 True / False
In a crab canon, the follower (comes) enters performing the same melody as the leader (dux) but transposed to a different pitch.
TTrue
FFalse
Answer: False
In a crab canon, the follower performs the leader's melody in RETROGRADE — backward from end to beginning. Pitch transposition is the defining feature of a standard canon (e.g., a canon at the fifth), but crab canons are defined by temporal retrograde, not pitch transposition. A mirror canon combines both retrograde AND inversion. Confusing crab canons with standard canons at a different pitch is a common error.
Question 4 True / False
A canon is fundamentally a compositional constraint — it derives an entire texture from a single melodic idea subjected to rule-governed transformation.
TTrue
FFalse
Answer: True
This is the central insight about canon as a structural form. Unlike free counterpoint where you can add whatever notes sound best, a canon commits the composer to deriving everything from the dux. The constraint simultaneously limits freedom (you can't insert a convenient note that isn't already in the melody) and generates coherence (the texture has an organic unity because it all comes from one source). This is why Bach and later composers like Bartók and Ligeti found canons so powerful: the constraint IS the compositional engine.
Question 5 Short Answer
Explain why understanding invertible counterpoint is a prerequisite for writing proportional (mensuration) canons.
Think about your answer, then reveal below.
Model answer: In a proportional canon, different voices perform the same melody simultaneously at different tempos. At any moment, the vertical interval between two voices is determined by which beats of the melody are sounding at the same time. Because voices move at different speeds, every possible combination of melody-points will eventually sound against every other. The melody must produce acceptable counterpoint with itself at ALL temporal offsets — which means every interval combination that arises must avoid forbidden parallels and dissonances. Invertible counterpoint is exactly the skill of writing lines that work regardless of which voice is on top; mensuration canons extend this to temporal displacement.
Josquin's mensuration canons in the 'L'homme armé' Mass work because his single melody, when played against a 2:1 or 3:1 slowed version of itself, never produces consecutive fifths or octaves. Achieving this is not luck — it requires systematic understanding of which intervals at which offsets remain legal. That is invertible counterpoint applied in the time dimension.