Triads with thirds that aren't major or minor?

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 Gf) or a dA (C Eff 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 Eff Gf) 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.

• Your dA (C Eff G) looks like a Csus2. – Scimonster Apr 16 '17 at 11:55
• Yep, I think all of these combinations can be understood enharmonically, I just didn't want to list them all. – Richard Apr 16 '17 at 11:55
• PS - It sounds like a Csus2 ;-) – Richard Apr 16 '17 at 11:56
• @alephzero But chromaticism at least theoretically introduces the possibility. Still not convinced it necessarily adds anything to our understanding, but augmented sixths aren't outside the realm of common-practice theory and are (rarely) written as diminished 3rds. – Pat Muchmore Apr 16 '17 at 15:39
• What I find more interesting are chords with third-like intervals from higher limits than the usual Ptolemaic 5-limit. One such interval is the septimal subminor third. – leftaroundabout Apr 16 '17 at 15:57

To my knowledge, no one has done this (outside of non-tonal theories like set theory). In fact, triadic harmony is generally defined as stacks of major and minor thirds only. As you point out in your question, most of the harmonies sound enharmonically identical to simpler harmonic ideas, and that too weakens any need for theorizing about or even using such spellings.

However, as you mention in a comment, enharmonic respelling does not in and of itself invalidate the possibility of a speculative theory for such structures. The main tonal example I can think of is the world of augmented sixth chords. If you were to naïvely try to stack up an It+6 as if it were triadic, then it appears to have a diminished third above the "root." For instance, in C major it would look like F#, Ab, C. The spelling indicates the function quite clearly, as the F#s job is to resolve up to the 5th while the Ab resolves down to it. The standard tonal theory response to this is that these harmonies actually aren't triadic at all and don't really have a root per se, but I suppose one could build from that to try to define a function for diminished 3rds in chord stacks.

Actually, there's a wealth of examples from +6 land. The German +6 sounds just like a dominant seventh chord but functions very differently. I think traditional theory already handles the harmony fine, but, again, one could stack it in such a way that it has a diminished 3rd and talk about its resolution in those terms. The Tristan chord sounds like a half diminished seventh, but its spelling indicates a different function. Et cetera.

Chords for the most part are built on a series of thirds. If you have for instance a chord with a diminished third, your third sounds no different than a Major second. This interval is way too dissonant to make a chord from. You need the strong intervals to make a chord.

As for a chord with a minor third and an augmented fifth and major third and diminished fifth. These chords will again not sound very good. The whole idea of a diminished chord is that both the third and th fifth yearn to be closer to the root, with the minor third and the dimindhed fifth go closer to the root. The opposite is true for a augmented chord both the major third and the augmented fifth go away from the root.

With these chords you mention you have a third that goes one way and a fifth that goes the other way. This lack of symmetry does not produce good sounds. Im sure there has been an investigation by some theorist in the past as to what would happen if you do this. I think that it simply does not sound good so it never became a thing.

Im not all that certain if you know exactly what it means for a chord to be dimished or augmented. A diminished chord has a minor third and a diminished fifth. There is no such thing as a minir fifth. A augmented chord has a major third and a augmented fifth.

• "As for a chord with a minor third and an augmented fifth...[it] will again not sound very good." But it sounds just like a major triad in first inversion, no? That sounds pretty good to me... "Im not all that certain if you know exactly what it means for a chord to be dimished or augmented." Obviously I do, since I explain them in my question. "There is no such thing as a minir fifth." I don't believe I ever said there was. – Richard Sep 18 '18 at 2:56