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The problem it solves
You often want to know which interval is left if you flip two notes, to choose voicings or to see why a fourth and a fifth are the same pair turned over. You need a quick rule instead of counting from scratch each time.
Detailed theory
Key idea
Rule of 9: the two numbers always sum to 9. 2nd↔7th, 3rd↔6th, 4th↔5th, unison↔octave.
The qualities swap: major↔minor, augmented↔diminished, and perfect↔perfect.
Understand it
Inverting an interval means turning it upside down: you take the lower note and raise it an octave (or, equivalently, lower the upper one an octave), so the one that was low becomes high and vice versa. The two notes are the same; you have only changed their vertical order.
There is a rule that makes this immediate: the rule of 9. The number of the original interval and that of its inversion always add up to nine. So a second inverts into a seventh (2+7=9), a third into a sixth (3+6=9), a fourth into a fifth (4+5=9) and the unison into an octave (1+8=9).
The qualities also have a rule of symmetry. On inverting, major and minor swap: a major third becomes a minor sixth. Augmented and diminished do the same: an augmented fourth inverts into a diminished fifth. And what is perfect stays perfect: a perfect fifth inverts into a perfect fourth.
Look at it with two examples. C–E is a major third (4 semitones); if you raise the C above the E, you get E–C, a minor sixth (8 semitones): 3 + 6 = 9, major↔minor, and 4 + 8 = 12, the whole octave. C–G is a perfect fifth (7 semitones); turned over it gives G–C, a perfect fourth (5 semitones): 5 + 4 = 9, perfect↔perfect. That is why a fourth and a fifth are really the same pair seen from two sides.
An analogy: it is like finding a number’s complement up to 9. If you have a 3, its partner is the 6; if you have a 4, it is the 5. Inverting an interval is exactly that: finding its complementary pair within the octave, by turning it upside down.
Staff & keyboard
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The complementary pair of the rule of 9: the third C–E and the sixth E–C'. Press the keys and notice they are the same notes with the order flipped.
How to recognise it
How it's written
There is no special symbol for inversion: it is written by swapping the order of the two notes on the staff, with the one that was low now on top. To name the result, subtract the original number from 9 (3rd → 6th) and change the quality for its complement (major → minor, aug → dim, perfect → perfect).
How it feels
On turning the interval over you hear the same pair of notes but with a different balance: the one that was below now crowns the interval. The complementary distance sounds related, like two sides of the same coin. Play C–E and then E–C to notice they share identity but change in width and colour.
Common mistake
Thinking the numbers add up to 8 instead of 9: a third inverts into a sixth (3+6=9), not into a fifth.
Forgetting to swap the quality: the inversion of a MAJOR third is a MINOR sixth, not a major sixth.
Try it
Play C–E (major 3rd) and then raise the C an octave to get E–C (minor 6th): confirm that 3 + 6 = 9 and that major has become minor.
Play C–G (perfect 5th) and turn it over to G–C (perfect 4th): you will see that perfect stays perfect and that 5 + 4 = 9.
On the instrument
Interval distance
C–E: a major third, four semitones. This is the interval we will invert.
Interval distance
E–C’: the inversion of C–E. Raise the C an octave and the major third becomes a minor sixth (8 semitones). 3 + 6 = 9, major↔minor.
Interval distance
C–G: a perfect fifth, seven semitones, before turning it over.
Interval distance
G–C’: the inversion of C–G. The perfect fifth becomes a perfect fourth (5 semitones). 5 + 4 = 9, and perfect stays perfect.
Where it's used
- Choosing voicings
- Turning an interval over to place the notes of a chord in a more comfortable or sonorous layout.
- Deducing intervals quickly
- Using the rule of 9 to name the inversion without counting semitones from scratch.
- Relating fourths and fifths
- Seeing why a fourth and a fifth are the same complementary pair turned over.
Examples
Interval distance
C–F#: an augmented fourth (6 semitones), the tritone, before turning it over.
Interval distance
F#–C': the inversion of the tritone. The augmented fourth becomes a diminished fifth (6 semitones). 4 + 5 = 9, aug↔dim.
Exercises
Practise interval inversions
Aural and visual recognition of intervals.
Complete 10 attempts · 70% accuracy to pass
Mini test
Check that you've got it.
0/8 answeredQuestion 1/8
What does it mean to invert an interval?