Inverting an interval by an octave produces its complement: numbers sum to 9, semitones to 12, and quality flips

To invert an interval, move its lower note up an octave (or the upper note down an octave); the result is the interval’s complement. Two invariants hold for any interval and its inversion: the two interval numbers always sum to 9 (second+seventh, third+sixth, fourth+fifth), and the two semitone counts always sum to 12 (one octave). Quality flips predictably: perfect stays perfect; major becomes minor and vice versa; augmented becomes diminished and vice versa. So a major third inverts to a minor sixth; a perfect fifth inverts to a perfect fourth; a minor seventh inverts to a major second. Note the two rules use different totals (9 for numbers, 12 for semitones) because interval numbers are one-based and count inclusively while semitones are a zero-based distance. Knowing these pairs builds interval-recognition speed and supports voice-leading analysis, since an inverted interval is just the same two pitch classes reordered across the octave.

Examples

Perfect fifth (7 st) inverts to perfect fourth (5 st): numbers 5+4=9, semitones 7+5=12. Major third (4 st) inverts to minor sixth (8 st): 3+6=9, 4+8=12. Minor seventh (10 st) inverts to major second (2 st): 7+2=9, 10+2=12.

Assessment

Without computing semitones, predict the inversion of (a) a minor third, (b) an augmented fourth, (c) a major sixth — giving both the number and the quality. Then verify each with both rules: numbers sum to 9 and semitones sum to 12.