Interval Inversions

Quick, what’s the interval between A and C?

Ok, it’s a trick question. The answer is it depends on which octaves the notes are in, and specifically which note is higher. That’s the concept of interval inversions that I will talk about here. I suggest reviewing my lessons on The Musical Alphabet and Intervals.

For reference, here is the table of interval names.

# of half-steps 1 2 3
interval name minor 2nd major 2nd minor 3rd
# of half-steps 4 5 6
interval name major 3rd perfect 4th augmented 4th/diminished 5th
# of half-steps 7 8 9
interval name perfect 5th augmented 5th/minor 6th major 6th
# of half-steps 10 11 12
interval name minor 7th major 7th octave

A good first guess answer to the above question would be minor third, since there are three half-steps moving upward from A to C (from A to A#, from A# to B, and from B to C).

But what if we take the A and raise it up an octave. Then we’re counting half-steps starting from C instead of from A. Like so:

  1. C to C#
  2. C# to D
  3. D to D#
  4. D# to E
  5. E to F
  6. F to F#
  7. F# to G
  8. G to G#
  9. G# to A

We can see that is nine half-steps. Or, if we’re clever (which we are), we can remember that an octave is 12 half-steps, so just subtract three from 12, and get nine that way. Either way, we see that the inversion of a minor third is a major sixth.

This process is called inverting an interval. That is, it’s when you take the lower note in an interval and increase it by an octave.

One important thing to notice is that if you invert an interval twice, you wind up with the same interval you started with. Using the above example, this becomes obvious. Start with the minor 3rd from A to C. First, raise the A an octave, giving a major 6th interval from C to the A above. Then raise the C an octave, and we’re back to the minor 3rd from A to C.

So what?

When dealing with harmonies, an interval and its inversion are like cousins, really close-knit cousins. Because they are really built from the same notes, but moved to different octaves, one can be substituted for each other. In isolation the we’re looking at them now, that’s not such a big deal, but use them to build chords, and it becomes a hugiferously important tool for making interesting harmonies. Next lesson, I’ll cover chord inversions, and you’ll see more clearly what I mean.

Special case: the tritone

If you’re good at math, you may notice that 12 - 6 = 6. Interesting, eh? That means that the interval of six half-steps is it’s own inversion. Checking the table, this is the awkwardly named diminished fifth / augmented fourth (those are actually two alternate names). It also has a third name, the tritone. It’s exactly half an octave, and actually not a particularly pleasant sounding interval. So much for cosmic balance.


Here is a listing of intervals and their inversions.

interval name minor 2nd major 2nd minor 3rd major 3rd perfect 4th augmented 4th/diminished 5th perfect 5th augmented 5th/minor 6th major 6th minor 7th major 7th octave
inversion major 7th minor 7th major 6th augmented 5th/minor 6th perfect 5th augmented 4th/diminished 5th perfect 4th major 3rd minor 3rd major 2nd minor 2nd octave