triadic inversion another way

Today's discussion is about inverting triads. What is generally meant when inverting triads is the following, done with C major:

That is to say, what we're really dealing with here is a re-ordering of the notes.

The concept of inversion as applied to melodic lines, however, has more to do with the actual meaning of inverting, i.e. turning upside down/placing in an opposite order (like a mirror). For example the following little line

will invert (diatonically, that is not adding any sharps or flats) to:

Now, what if we apply the same idea to chords? Something interesting will happen. We'll invert C major three times, first with C as the axis of symmetry, or mirror line:

The chord we end up with is an F major chord. Now let's use G as the axis:

Now we've produced a G major triad. So the interesting point here is that simply by inverting a triad (let's say melodically) we end up with the IV (subdominant) and V (dominant) chords. Not only that, these inversions have given us all the notes of the key of C major.

Oh, and the last way to invert the triad, with E as axis, produces...

...yeah, we get the same chord right back.

These are, by the way, diatonic inversions. Next time we'll examine what happens when we invert our intervals strictly.