I don't even know why I'm asking this question, but here we go:
The four common triads are built with different combinations of a major and/or a minor third:
- A major third on the bottom with a minor third on top (let's call this Mm) produces a major triad.
- mM creates a minor triad.
- mm creates a diminished triad.
- And MM creates an augmented triad.
Have any theorists ever developed a system where triads could be built using augmented or diminished thirds?
For instance, has anyone ever discussed the function of a Md (C E G♭) or a dA (C E♭♭ G) triad?
I'm specifically looking for a tonal approach to this question; obviously pitch-class sets can explain constructions like this, but that's not what I'm going for.
I'm also not looking for enharmonic approaches to this. For instance:
- dM (C E♭♭ G♭) is enharmonic to an incomplete D7 chord,
- AA (C E♯ G♯♯♯) is an enharmonic quartal chord,
- and MA (C E G♯♯) is just a minor triad.
This is not what I'm looking for, but it may be that the enharmonic equivalence ultimately resulted in theorists deciding that it's just not worth the effort to theorize these types of triads. That's okay too!
Obviously this would be much more of a "speculative" theory, meaning that it's unlikely we would see it in "real" music very often. But the idea crossed my mind today, and I thought I'd ask.