The rules for inverting intervals are well documented (for example on Wikipedia), but I'm having a hard time figuring out why this operation is of interest:
How does interval inversion help us in understanding or creating music?
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In addition to Dom's great answer, there's a very famous compositional technique known as invertible counterpoint. In short, invertible counterpoint switches the register of two (or more) independent voices such that the voice that was on top is now on the bottom and the voice that was on bottom is now on top.
Now let's remember that, from a standpoint of strict counterpoint, a perfect fifth is consonant and a perfect fourth is dissonant. As such we need to have an awareness of how intervals invert so that if we write a counterpoint that will eventually be inverted, we make sure to treat particular intervals carefully so that their inverted versions also follow all rules of consonance and dissonance.
The simple explanation is that inverse intervals are another way to view the distance starting at a different point You can go up to a note or down to a note which gives different interval sizes, but they are related. Where you see this concept most is looking at vertical harmony especially chords.
To keep it simple, we'll just look at the notes C and G. When you are talking about intervals the default is to assume upward and C up to G would be a perfect 5th. However when looking at C down to G, the interval would only be a perfect 4th. The distance is different because we are taking a different path. This is also why G up to C would be a perfect 4th and G down to C would be a perfect 5th.
Again if you looked at a harmonic structure containing C G and C like below, you would have both intervals and on the staff you could see the difference in staff space between the inverse intervals.
X:1 L:1/1 M: K:C V:1 clef=treble [C G] | [G c] | [C G c] %
@Dom and @Richard gave great answers. In fact you can sort of combine them and look into imitative counterpoint and fugue where the "entries" of imitative counterpoint can occur at the fifth above and the fourth below. The musical importance of the voices entering at those points of inversion above and below is it keeps clear what key the music is in by highlighting the tonic/dominant relationship.
I can add another musical aspect of inversion. Some inversions procedures can sort of "flip" musical qualities.
C major chord is a major third (M3) and a minor third (m3) above a root
C M3↑ E m3↑ G.
We can invert that. But first we need a point from which to invert - this is an important aspect of inverting, inversion can happen from different points.
C major chord from the
C root we will invert/reverse the added intervals - add a M3 below and then a m3 below the root
C M3↓ Ab m3↓ F
If we start on
G the inversion give us a
A similar thing happens if you invert the whole and half step direction of a major scale starting on
G. It will produce the minor scale...
Some chord are symmetrical when inverted. Ex. if you invert a diminished seventh chord at the root, you get the same chord, enharmonically respelled...
Other chords change quality when inverted, like a dominant-seventh chord from a major key inverted at the tonic becomes a half-diminished seventh chord from the parallel minor key...
...help us in understanding or creating music?
I'm not sure if awareness of these inversions helps much. Apparently they figure large in Riemannian theory. Personally I don't buy into the idea that the minor scale is derived from a major scale inverted at the fifth. I don't think that fits into any acoustical phenomena nor what I understand of ancient music theory & history. But these inversion can be observed - if not necessarily heard - and they come up in music theory discussions from time to time.