---
title: Imitation and Invertible Counterpoint
dek: Imitation makes one idea serve two lines. Invertible counterpoint makes those lines interchangeable, and the arithmetic behind it is small enough to check.
topic: counterpoint
tags: [imitation, canon, invertible counterpoint]
date: 2026-09-25
author: Yunus Emre Vurgun
level: advanced
related: [first-species-counterpoint, fugue-subject-and-answer, voice-leading-in-four-parts]
---

Two melodies that have nothing to do with each other can still be written together correctly, and the result will often sound like two people talking past each other. Imitation is the oldest solution to that problem: the second voice takes up the first voice's idea, and the ear immediately hears a relationship that no amount of correct part-writing can manufacture.

Invertible counterpoint goes one step further and asks what happens if the two voices trade places. It sounds like a compositional parlour trick, and in the sixteenth century it often was one: a demonstration that a piece of counterpoint is so well made that it survives being turned upside down. But the technique is also the mechanism behind fugue subjects that combine with their countersubjects in any order, and behind development sections that shuffle the same two ideas through every register the orchestra owns.

## Imitation: one idea, two entries

In its simplest form, one voice states a melodic idea and another voice enters with the same idea, usually at a different pitch level. The entry comes after the first voice has stated enough of the idea for the listener to recognize it: typically the length of the motive, a beat or a bar, sometimes longer.

Everything about the device depends on that entry interval. Entering at the fifth or the octave is the norm in tonal music because those intervals sit inside the tonic-dominant relationship that defines the key; an entry at the fifth announces a world with a tonic and a dominant in it. Entries at the second, third, sixth or seventh are common in later and modal repertoires, but they do not articulate a key in the same way, so they tend to read as colour rather than as an answer.

Imitation can be strict or free. Strict imitation repeats the motive exactly, interval for interval, transposed. Free imitation keeps the rhythm and contour but alters the intervals. Between those two extremes lies the adjustment that tonal music needs constantly: **tonal imitation**, in which a motive is transposed only as far as the key allows. A motive that begins on the tonic and leaps to the dominant has to be treated carefully when it enters on the dominant, because a literal transposition would leap to the supertonic and change what the idea means harmonically. The traditional fix shortens the opening leap of a fifth to a fourth, which is exactly the adjustment that produces a tonal answer in fugue.

## Canon as the limit case

Canon is imitation with no exit: the following voice repeats the leader strictly, from the beginning to the end, so that every note of the piece is part of the imitative relationship. The simplest kind is a round, where the second voice enters after a fixed delay at the unison or octave and everyone can sing the whole melody.

The technique gets richer when the entry is transposed or transformed. A canon at the fifth enters a fifth higher; a canon by inversion flips the leader's intervals upside down, so that an ascending third in the leader becomes a descending third in the follower; a canon by retrograde plays the melody backwards. Mensuration canons change the speed of the following voice instead of its pitch, so the same notes are read in a different proportion.

The Goldberg Variations show how far the device can be organized across a large work. Between the variations, Bach places canons at successively larger intervals, from the unison through the second, third, fourth, fifth, sixth, seventh and octave to the ninth, so that the whole set is a graded anthology of what imitation can do at each entry distance. The point is not display for its own sake. Writing a canon at the ninth is a genuine problem in register and interval choice, and the solutions are the same solutions a fugue needs.

## Inverting the texture

Invertible counterpoint asks you to write two lines so that either one may be the upper voice. The lower voice may be moved above the upper (typically at the octave, the tenth or the twelfth), and both arrangements must still be consonant.

The arithmetic that governs this is simple once you see it. When two voices are inverted at the octave, an interval and its inversion add up to nine: a second becomes a seventh, a third becomes a sixth, a fifth becomes a fourth, an octave becomes a unison. The list is worth memorizing because it tells you immediately which intervals survive the flip.

| Interval above | Inverts to | Usable in two-voice writing? |
| --- | --- | --- |
| Unison | Octave | yes |
| Second | Seventh | no; dissonant both ways |
| Third | Sixth | yes; the safest interval available |
| Fourth | Fifth | no; the fourth is dissonant in two voices |
| Fifth | Fourth | no; see above |
| Sixth | Third | yes |
| Seventh | Second | no |
| Octave | Unison | yes |

The fourth is where this bites. In two-voice writing a perfect fifth above the lower voice inverts into a perfect fourth above the lower voice, and a fourth over the bass is a dissonance. So a strict two-voice exercise invertible at the octave cannot use fifths at all, which is a severe restriction: thirds and sixths become the entire vocabulary. Add a third voice and the restriction relaxes, since a fourth above the bass can be consonant if one of the upper voices supplies a fifth or a third above it. This is why invertible counterpoint is taught first in two voices and used mostly in three and four.

Quality inverts along with size. A major interval becomes minor when the voices swap; perfect intervals stay perfect; augmented and diminished trade places. So a major third above the lower voice comes back as a minor sixth below it, which is consonant in both directions and therefore perfectly usable, even though it does not sound identical.

Here is a pair of parallel thirds, then the same four pitch classes with the voices swapped:

[keys]C4+E4 D4+F4 E4+G4 F4+A4[/keys]

The interval above the lower voice is a major or minor third at every step. Now the same material inverted at the octave, with the formerly lower line on top:

[keys]E3+C4 F3+D4 G3+E4 A3+F4[/keys]

Every vertical interval is now a third's inversion, a sixth, and the counterpoint still works. The pitch content is identical in the two versions; only the register, and therefore which line is on top, has changed.

## At the tenth and the twelfth

Inverting at the octave is not the only option, and composers often prefer a different interval because it permits intervals the octave forbids. At the tenth, an interval and its inversion add up to eleven: a third inverts to an octave, a fifth to a sixth, a sixth to a fifth, and an octave to a third. Thirds, fifths, sixths, octaves and tenths are all usable, which is a much larger palette than the octave allows.

At the twelfth, the sum is thirteen: a third inverts to a tenth, a fifth to an octave, an octave to a fifth, and a tenth to a third. The sixteenth-century writers valued this arrangement because a fifth between the voices becomes an octave when the texture is flipped, a strong interval in both positions. But a sixth inverts to a seventh, so sixths are unavailable in strict two-voice writing invertible at the twelfth.

The rule generalizes cleanly: an interval and its inversion must both be consonant. That is the whole discipline. Everything else, including which inversion interval to choose and whether the flip happens at a phrase boundary or in the middle of a line, is a compositional decision about where you want the strong intervals to land.

## Where it shows up, and where the idea fails

Fugue is the form that lives on invertible counterpoint. A subject and its countersubject are written so that they can be combined in either order, at the octave or the twelfth, in whatever key the middle entries require; the exposition's whole architecture depends on that property holding.

Development sections use the same mechanism on a larger scale. Haydn and Beethoven frequently take two ideas from an exposition and combine them in invertible counterpoint, which is a way of proving that the two themes belong to the same piece. The coda of the finale of Mozart's Symphony No. 41 does this with several distinct figures at once, stacked so that every voice is audible and none of them cancels another.

Two misunderstandings are worth clearing up. First, imitation is not the same as sequence. A sequence repeats a motive in one voice at successively higher or lower pitch levels; imitation requires a second voice to take up the idea, and the two devices have different effects even when they use the same motive. Second, inversion in this sense has nothing to do with chord inversion or with melodic inversion by mirroring. Invertible counterpoint means swapping registers, not flipping intervals.

The deeper caution is about writing invertible counterpoint where it will never be inverted. The technique imposes real costs, in restricted intervals and careful register planning, and paying those costs for a passage that will never trade places is wasted effort. It is worth doing when the material is going to combine in several orders, and it is worth doing as an exercise because it improves how you hear the independence of two lines, and for no other reason.

## Working with it

Write the two lines as one piece, not as a melody and an accompaniment, then test the flip before you like either version. If the lower voice's line sounds like an accompaniment when it moves on top, it was an accompaniment. Play both versions on the [virtual piano](/tools/virtual-piano/), and check the interval at every arrival rather than checking once at the end.

Then set yourself a problem that forces the issue: take an eight-note idea, write a line against it, invert the pair at the octave, and repair whatever broke. Most first attempts break in the same two places: a fifth that inverts to a fourth, and a climax in the upper voice that becomes a high note buried in the middle of the texture when the parts swap. Both are register problems, and both are solved the same way, by deciding where the two lines are allowed to travel before writing a single note of either.