If you look at the kinds of thirds and fifths, you immediately see six obvious combinations:
- minor third/diminished fifth
- minor third/perfect fifth
- minor third/augmented fifth
- major third/diminished fifth
- major third/perfect fifth
- major third/augmented fifth
Four of these comprise the complete set of so-called triads:
- m3+d5 = a diminished triad
- m3+P5 = a minor triad
- M3+P5 = a major triad
- M3+A5 = an augmented triad
Of the two which remain, the minor third/augmented fifth combination results in another major triad in 1st inversion. However, the last remaining combination—major third/diminished fifth—is simply ignored, overlooked, or perhaps purposely excluded from triadhood. Why?
I'm sure someone is going to think, "Ah, but triads are constructed by stacking thirds, and the distance from a major third to a diminished fifth is a major second."
And I would say, "Well, if you look at an example like C-E-G♭, it seems to me that E to G♭ is a diminished third, not a major second."
♭5 chords are a thing, particularly in the "triadic" portion of dominant chords, so I must guess that there is some historical reason why they aren't considered triads. (Perhaps because classical theory was well-established before jazz came about?) What is it? Or am I missing something?