Interval inversion

Take C and F, with the F above. It is a perfect fourth. Now take the C up an octave: the same two notes, F and C, only the C is now on top. The interval has changed, and it is now a perfect fifth.

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That is the inversion of the interval: the two notes swap positions, and the lower one becomes the upper (Lacerda 1966, ch. XXXIII). There are two ways to make the swap, and both give the same interval: take the lower note up an octave, or the upper note down an octave.

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The rule of nine

The interval's number changes in a fixed way. A fourth becomes a fifth, a third a sixth, a second a seventh, a unison an octave, and back again the same way (Lacerda 1966, ch. XXXIII):

Interval	Inverted
unison	octave
second	seventh
third	sixth
fourth	fifth
fifth	fourth
sixth	third
seventh	second
octave	unison

An interval and its inversion always add up to nine. To find the inversion, take the number from nine: the inversion of a second is 9 − 2 = 7, a seventh (Med 1996, ch. XIII).

Why nine, when together the two intervals cover an octave? For the same reason as the sum for compound intervals. From C up to F is four names; from F up to C, five; and F is counted in both. Four plus five is nine, but the distinct names are only eight.

Quality swaps too

Quality changes along with the number, and always to its opposite (Lacerda 1966, ch. XXXIII):

Interval	Inverted
major	minor
minor	major
augmented	diminished
diminished	augmented
perfect	perfect

Doubled qualities follow the same logic: doubly augmented becomes doubly diminished, and the reverse. And perfect is the only one that stays as it is.

Counting semitones makes the reason clear. An interval and its inversion, together, fill exactly one octave, twelve semitones. If one is large, the other has to be small. A major third has four semitones; what is left of the octave is eight, a minor sixth.

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A few more, to watch the rule work:

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E–G is a minor third, and G–E a major sixth. C–D is a major second, and D–C a minor seventh. F–B is an augmented fourth, the tritone, and B–F a diminished fifth. Blatter reaches the same table by going through examples like these, one pair at a time (Blatter 2007, part 3, art. 1).

What it is good for

Inverting is a shortcut for measuring intervals downwards. Blatter gives the example: how far is it from the tonic down to the note a fourth below? Instead of counting again, invert. If C up to the F above is a perfect fourth, C down to the F below is its inversion, a perfect fifth (Blatter 2007, part 3, art. 1).

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The same helps when reading a chord whose notes are stacked in an unexpected order: knowing that a minor sixth is a major third upside down saves the count.

Where the books differ

Med makes a distinction the other two do not. In a melodic interval, inversion keeps the order of the notes: the first note is still the first, and the interval only changes direction, from ascending to descending. If the order of the notes changes, Med does not call the result an inversion but the interval's octave complement (Med 1996, ch. XIII). In a harmonic interval, where the notes sound together, the question does not arise.

Med also warns that compound intervals are generally not inverted. Taking the lower note of a tenth up an octave only gives back the corresponding third, not an inversion. Some theorists do invert them in steps: reduce to the simple interval, invert that, and add the octaves back, so a tenth becomes a thirteenth.

What comes next

Some inversions sound close to their original, others quite different. Perfect fourths and fifths sound stable; seconds and sevenths, tense. That difference between rest and tension is the subject of consonance and dissonance.