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The third of the chord is one of the criteria to determine the name of the chord. If the third is major (and the fifth is perfect), the chord is called major chord ; if the third is minor and the fifth is perfect, the chord is called minor chord ; if the third is minor and the fifth is diminished, the chord is called diminished chord.
But what do you call a chord that has a diminished third?
Like, what kind of D chord would D#,F,A be?
The answer is actually quite simple: it's called a double-diminished triad. Played in first inversion it's usually referred to as Italian augmented sixth chord. As mentioned in other answers, our ears have a tendency to hear it as a dominant seventh sound.
This is a tough question. [0,2,6] probably isn't what you're looking for, but I believe is the best way to address it. It's going to sound like an F7(4 2), but is certainly not written that way.
It could resolve to E, as F7 can be a tritone substitution for B7 in terms of function. To support this, both D# and F could resolve to E (upward and downward, respectively), and the A to G#. And the D# is written with a sharp, suggesting that it would continue with upward motion.
In my career of theory, I've never heard of a qualitative term for a triad that begins with a diminished third. But it's a good question.
Not every set of notes yields a nice, clean chord name just stacking in thirds. Sometimes it is necessary to rearrange the notes to see how they fit better especially since there are no chords that are contain a diminished 3rd. If you rearrange the letters like so:
F A _ D#/Eb
You get and F7 like others have stated. The full name would be F/D#.
If you really wanted, you could call it a D#sus2b5 to denote a diminished 3rd as a sus2 and just denote the b5.
A D# chord like this could never occur in any scale degree of a major or minor scale with the d# note as a root so there would be no need to name it with reference to the d#.
However this notes could occur together, for example in a B7♭5 chord without the root (b).
Since music theory doesn't care for something that could only happen outside a harmonic context, at least with what is broadly accepted as harmonic, there is no actual way to accurately represent this phenomenon with a standard name.
I would interpret this as an F7 chord with the 5 missing. If you respell the D# as Eb, it will make more sense. In my experience I have never encountered a diminished third as a definitive chord voicing.
By analogy with Why does the C7 chord on guitar omit the G note (5th) in open position? this chord could perfectly acceptably be named as an F7 (no 5th.) As others have said, Eb F A is going to sound like an F7, and the chord shape discussed in the question I have linked shows there is indeed a precedence set for naming chords like this as 7ths.
Many guitarists will have put a capo on a guitar at the 5th fret, fingered the shape from the linked question, played Eb F A and called it F7 without even thinking about it. As discussed in the answers to linked question, the 5th is the most expendable note in the chord.
It'll SOUND like a dominant 7th , although technically it won't be called that. As we are aware, the 5th of a chord is one that can be left out.
