Interval Inversion · Jazz Wiki

Interval inversion is what happens when you flip an interval upside down — take the bottom note and put it on top (or the top note underneath), usually by moving it an octave. It sounds like a simple trick of arithmetic, but it explains why a major 3rd and a minor 6th are secretly the same relationship viewed from opposite ends, why two dominant chords a tritone apart can substitute for each other, and why stacking fourths gives you the same notes as stacking fifths. Once you see the pattern, whole chunks of jazz harmony — Tritone Substitution, Quartal Harmony, Rootless Voicings — stop looking like separate tricks and start looking like one idea applied in different places.

The rule of 9 and why it works

Every interval and its inversion add up to 9: a 3rd inverts to a 6th (3+6=9), a 5th inverts to a 4th (5+4=9), a 2nd inverts to a 7th (2+7=9), a unison inverts to an octave (1+8=9). This isn’t a coincidence — it’s because both notes of the original interval are still present in the inverted pair, and the octave itself always divides into two complementary spans. Think of it like a seesaw: however far one note sits from the bottom of the octave, the other note fills in the rest, and those two distances always sum to the whole span plus one (because we count endpoints twice in interval-counting).

Quality flips, but perfect stays perfect

Interval quality doesn’t survive inversion unchanged — except for the perfect intervals, which are the stable ones:

Major inverts to minor, and minor inverts to major (M3 ↔ m6, M6 ↔ m3, M7 ↔ m2, M2 ↔ m7)
Perfect stays perfect (P4 ↔ P5, P1 ↔ P8)
Augmented inverts to diminished, and vice versa (aug 4 ↔ dim 5)

In C major: C–E is a major 3rd; flip it and E–C is a minor 6th. C–G is a perfect 5th; flip it and G–C is a perfect 4th — same quality, because perfect intervals are the symmetrical center of the system. C–F♯, an augmented 4th (the tritone), inverts to F♯–C, a diminished 5th — different names, same six half steps, and by enharmonic equivalence they sound identical no matter which way you spell them.

Why the tritone is the odd one out

The tritone is the only interval that inverts to itself in sound: six half-steps up is the same distance as six half-steps down within the octave, so aug 4 and dim 5 are enharmonic mirror images of each other. That symmetry is the entire engine behind Tritone Substitution. In a G7 chord, the 3rd and 7th are B and F — a tritone. In D♭7, the 3rd and 7th are F and C♭ (spelled enharmonically as B) — the exact same tritone, just relabeled. Because inverting the tritone gives you back a tritone, G7 and D♭7 share their most identifying pitches and can substitute for one another in a ii–V–I, often producing a smooth chromatic bass line into the tonic.

Fourths as inverted fifths

Stack perfect 5ths — C, G, D, A — and you get the skeleton of traditional tonal harmony. Invert each of those 5ths into a 4th and restack them as C–F–B♭–E♭, and you get quartal voicings: the same underlying interval relationship, turned inside out. This is why quartal harmony sounds modal and open rather than functional — it deliberately avoids the 3rds and 7ths that usually tell your ear “major” or “minor,” leaning instead on the ambiguous, stacked-4th sound. Bill Evans’ famous voicing on So What (E–A–D–G–B over a D bass) is built this way, and The So What Voicing became a template for modal comping. Herbie Hancock uses the same fourths-based language on Maiden Voyage, and McCoy Tyner extends it into a percussive, cascading style.

Here is that voicing — stacked perfect 4ths (E–A–D–G) topped by a major 3rd (G–B), over a D bass:

Interval inversion also quietly powers Drop 2 Voicings and Rootless Voicings: when you drop a note down an octave or omit the root, you’re inverting intervals within the chord to make it fit under the hand while keeping the same pitch classes. That’s a close cousin of, but not identical to, Chord Inversions, which reorders a whole chord’s notes rather than a single interval — worth keeping straight, since the two get confused often. Good Voice Leading between chords often comes down to choosing whether to move a top voice or a bottom voice, and that choice is really a choice between an interval and its inversion.

♫ Listen

Miles Davis — “So What” (Kind of Blue, 1959): Bill Evans’ opening five-note voicing, E–A–D–G–B, is stacked 4ths — inverted 5ths — under Paul Chambers’ bass statement of the D Dorian tune. Listen for how it sounds neither major nor minor.
Herbie Hancock — “Maiden Voyage” (Maiden Voyage, 1965): the comping voicings stack perfect 4ths over a modal bass, giving the tune its open, “oceanic” quality that quartal harmony is known for.