Inverted and Compound Intervals
about 13 min
What this lesson is for
Flip an interval upside down and the two numbers always sum to nine, while major swaps with minor and augmented with diminished. There is a reason for that rule, and it also explains why perfect intervals are called perfect. Compound intervals beyond the octave round out the lesson.
- Explain why an inverted interval's number is found by subtracting from nine
- Explain why inversion swaps major with minor and augmented with diminished, but leaves perfect unchanged
- Reduce a compound interval to its simple form and name it
- Use inversion to halve the number of intervals you have to memorize
In this lesson
The previous lesson fixed an interval as a number plus a quality. This one asks what happens when you turn one upside down.
It can look like a piece of arithmetic trivia. It is not: chord inversions — first inversion, second inversion — are this same operation applied to a stack of intervals rather than a pair.
Inversion means swapping which note is on top
To invert an interval, raise the lower note by an octave (or drop the upper note by one). The two notes trade places.
Invert C–E, a major 3rd, and C moves up an octave to give E–C.
The set of note names has not changed — it is still C and E. Yet the interval name goes from major 3rd to minor 6th. Both halves of the name changed.
The two numbers always sum to nine
The rule for the number is simple.
Original number + inverted number = 9| Original | Inverted |
|---|---|
| unison | octave |
| 2nd | 7th |
| 3rd | 6th |
| 4th | 5th |
| 5th | 4th |
| 6th | 3rd |
| 7th | 2nd |
| octave | unison |
Why nine and not eight? Because of inclusive counting, from the previous lesson.
Put two notes inside one octave. Call the distance from the lower note to the upper one a, and from the upper note to the next octave b. Both spans count the middle note, so it gets counted twice, and a + b comes to 8 + 1 = 9. The number falls straight out of the counting convention. There is nothing here to memorize.
Check
Invert a minor 7th. What number results?
Qualities swap — except perfect
Qualities change under inversion just as predictably.
| Original quality | Inverted |
|---|---|
| major | minor |
| minor | major |
| augmented | diminished |
| diminished | augmented |
| perfect | perfect |
Half steps make the reason visible. An octave is twelve half steps, so an interval of n half steps inverts to 12 − n.
Major 3rd (4 half steps) → 12 − 4 = 8 half steps = minor 6th ✓ Perfect 5th (7) → 12 − 7 = 5 = perfect 4th ✓ Augmented 4th (6) → 12 − 6 = 6 = diminished 5th ✓
And here is where the perfect intervals earn their name. The perfect unison (0) and perfect octave (12) invert into each other, as do the perfect 4th (5) and perfect 5th (7) — and in both pairs both members are perfect. That closed symmetry across exactly four numbers is why 1, 4, 5, and 8 needed a vocabulary of their own.
The major/minor family has no such symmetry, since inversion flips major to minor every time. That asymmetry is precisely what the second set of terms exists to describe.
What inversion does to the sound
Check: The first is the major 3rd C–E; the second is its inversion, the minor 6th E–C. Same two letter names, noticeably different character. The third and fourth are the perfect 5th and its inversion the perfect 4th, where the character barely shifts — both stay solid and open.
four half steps
Check
What does an augmented 2nd become when inverted?
Inversion halves what you have to memorize
The practical payoff of all this is that you only have to learn half the intervals.
Plenty of people struggle to recognize a major 6th by ear while finding minor 3rds relatively easy. Since a major 6th is the inversion of a minor 3rd, it can be re-heard as “a minor 3rd turned over.” The same trick works across the board:
- major 7th ⇄ minor 2nd (the half-step clash)
- minor 7th ⇄ major 2nd
- minor 6th ⇄ major 3rd
- perfect 4th ⇄ perfect 5th
Which means that getting the four 2nds and 3rds solid gives you the 6ths and 7ths by inference. For ear training specifically, that is a far more efficient order of attack than working through all twelve independently.
Compound intervals — beyond the octave
Everything so far has stayed inside one octave. Those are simple intervals; anything larger is compound.
Counting works exactly as before: include both ends and count letters. C4 to E5 counts ten, so it is a 10th.
To go the other way, subtract seven to reduce a compound interval to a simple one — that removes one octave. Note that it is seven, not nine; nine was for inversion, which is a different operation.
- 9th → 2nd
- 10th → 3rd
- 11th → 4th
- 13th → 6th
The quality survives reduction untouched. A major 10th reduces to a major 3rd; a perfect 11th to a perfect 4th.
Those numbers 9, 11, and 13 are exactly the ones that appear in popular chord symbols. The digits in C9, Cadd9, C11, and C13 are compound interval numbers. Why write 9th rather than 2nd — because the note really is stacked an octave up rather than sitting next to the root, and the unit on extended chords later returns to what that notation is claiming.
Check
Reduce a perfect 12th to a simple interval. What is it?
Interval inversion and chord inversion are the same idea
One preview of where this is going.
A C major chord is C–E–G: a major 3rd with a minor 3rd stacked above it. Move the bass to E — first inversion — and it becomes E–G–C, which is a minor 3rd with a perfect 4th above it.
Describing that change in structure is precisely what interval inversion does. A chord inversion is a bundle of interval inversions, and Unit 3 takes it up directly.
Check an inversion in the interval calculator Read off C–A as a major 6th, then set the lower note to A and the upper to C. You get a minor 3rd, and the two numbers sum to nine Interval Calculator → Use inversion to lighten the ear-training load Get minor and major 3rds solid before you touch 6ths. Hearing a 6th as an upside-down 3rd makes it suddenly much easier to identify Ear Training →Exercises
Every answer here is cross-checked against the theory engine. Work them out before you grade.
- 1
What does a major 3rd (C–E) become when inverted?
- 2
What does a perfect 5th (C–G) become when inverted?
- 3
What does an augmented 4th (F–B) become when inverted?
- 4
What is the interval from C4 up to E5?
- 5
Reduce a major 9th to its simple interval. What is it?
Before you move on
Check that you can explain each point above in your own words.