It only dawned on me recently, but it's quite a fundamental and important feature of scales that they are not mirrored under their own inversion.
The intervalic formula for major is:
W W H W W W H
If we invert this, we end up with phrygian:
H W W W H W W
But this actually matters a lot, because if melodies in a given key are descending, they have different intervals than if they're ascending. I noticed this when writing a song in pentatonic major, where I kept using a descending minor sixth (my favourite interval). Any time the melody was descending, I was using really fun, spicy intervals that were distinctly non-major. So for example, consider the ascending melody in C major pentatonic:
C E A C. This has intervals above the tonic of major third, major sixth, octave.
But if we had some descending melody like:
C E D C ,
the intervals below the tonic are: minor sixth, minor seventh, octave.
But it's not just in melody. Whenever chords are voiced like:
1 3 8, you can hear the very distinct minor sixth interval. In other words, when a chord is voiced in such a way that the root is played in a higher octave, the other notes in the chord have these intriguing "descending interval" relationships with it.
With all that said, I never read about this when reading up on music theory. I Googled it and the term "scale inversion" just takes you to investment websites. The closest concept in music theory I could find is "melodic inversion", but that discusses the opposite case - keeping intervals unchanged under inversion.
Is there a reason why this seemingly important property of scales is not considered in music theory?
My guess at an answer, after discussing with someone: the psychoacoustics of harmony is that we consider the root of the chord to, well, root the chord. The chord's character is largely determined by the root. And in the cases when we do things like invert the chord in funny ways, we create ambiguity about what the chord is. For example, if we take the second inversion of C major:
G E C, we might hear it as C major, but we'll also kind of hear it like a Gsus4.
So to summarise my question: wow, scale inversions are really important, why is nobody talking about them?