MusicaLexica

An inversion is when you keep the same note names but flip the registers of the notes.  For example, C - E is a major third.  E - C is the same note names but now we’re travelling a longer distance…in fact, it’s a minor sixth.

Perfect intervals remain perfect intervals.  Unison remains unison, and an octave remains an octave.  A perfect fifth, when inverted, becomes a perfect fourth, and vice versa.

Minor intervals become major intervals and major intervals become minor intervals when inverted.

The intervals invert according to this chart:

second	<–>	seventh
third	<–>	sixth
fourth	<–>	fifth

So, a minor second becomes a major seventh.  A major second becomes a minor seventh.  A major third becomes a minor sixth, etc.

It can be easier sometimes to see the qualities of smaller intervals.  Is Bb - A a major seventh or a minor seventh?  Well, A - Bb is a half-step, which is a minor second.  So Bb - A is a major seventh.

Once you know the chords built with natural notes, figuring out thirds is pretty easy.  D - F is a minor third, so F - D is a major sixth. 

How about Ab - F?  We know F - A is a major third, so F - Ab is a minor third.  So Ab - F is a major sixth.

Of the natural notes, the only fifth that is not a perfect fifth is the interval from B - F, which is a diminished fifth.  (By the way, a diminished fifth inverted is another diminished fifth–or an augmented fourth, depending on how you look at it.  But the distance is the same.)

Any of the rest of the fifths made with natural notes are perfect fifths, and invert to perfect fourths.  G - D is a perfect fifth, D - G is a perfect fourth.  E - B is a perfect fifth, B - E is a perfect fourth.