Complementary interval
The interval that completes a given one to an octave — the fourth and the fifth, for instance.
What it means
The complementary interval is that interval which completes a given one to an octave. Fourth and fifth complement each other, as do third and sixth, second and seventh, prime and octave. It is thus the same as the result of interval inversion, viewed from another side: there a note is displaced, here one asks for the missing size.
Musical use
The rule is simple and holds without exception: the numbers of two complementary intervals add up to nine, and their designation reverses — major becomes minor, augmented becomes diminished, perfect remains perfect. A major third completes to a minor sixth, a perfect fourth to a perfect fifth, an augmented fourth to a diminished fifth.
Practically this is useful for several reasons. When singing, an uncomfortable wide leap can be derived by way of its complement: a sixth upward corresponds to a third downward plus an octave. When determining in the notation, a large interval can be reduced to its smaller counterpart. And in the study of composition, the relation explains why certain intervals are evaluated together: third and sixth are both imperfect consonances, second and seventh both dissonances.
In counterpoint, the double leading of two voices rests on the same principle: if the voices are exchanged, every interval is transformed into its complement — which is why fifths become fourths in the process and the passage must be checked anew.
The sum of two complementary intervals always yields an octave, their degree numbers add up to nine. Complementary intervals arise through inversion. Quality and sonic relatedness stand in a fixed relation: perfect intervals stay perfect, major becomes minor, augmented becomes diminished. In analysis this connection is helpful, because inverted intervals can be treated as equivalent.
Performance practice
Use the rule in ear training: practise every interval together with its complement, upward and downward. With difficult leaps, think the complement in the opposite direction — this hits more securely than trying to hear the wide distance directly.