Is Ab-D-Eb a sus4 chord?
Say I'm in C minor and want to use a VIsus4 chord. Do I use D as my sus4 note? I tried this and got a great, cinematic sound but I'm not sure it's called a sus4 as the interval is not a perfect fourth. Is it?
It seems to my ears that I can use this "augmented sus4" in my chord progs in the same way as a real sus4. Is this true?
Also, I tried using a Db instead of the D and it sounded weird. Maybe because it's not diatonic?
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There is an idea of nomenclatures, disseminated by Rick Beato, that generalizes the names of the triads. In this case, 1-?4-5 would be a "lydian triad" (as seen in this video at 2:42 Why Triads Are: The Building Blocks of Harmony). A "phrygian triad" would be 1-?2-5. These 2 triads along with major, minor, sus2 and sus4 form all the possible combination with a perfect fifth. Then the same process can occur with a diminished fifth and an augmented fifth (which has 2 combinations that are simply inversions of minor and major triads), generating names and symbols like C?5, Csus4(?5), C locrian (1-?2-?5), etc...
Your procedure is the correct way to generate suspended chords inside the scale: raise the middle note one degree using the notes of the scale (or lower it). You're also right about the 4th. A sus4 chord has a perfect fourth by definition. I'd like to call it an "augmented sus4" too, but unfortunately that's not an established name, although logical. If you really, really want to communicate just these 3 notes in a written scenario, maybe you could use the horrible symbol A?add?4no3 (or A?add?4omit3). If the 3rd is harmless to you and is even played by some instrument, you could say it's an A?add?4. If you have reasons to believe there's a "hidden" root somewhere else other than the A?, it could become an E?Maj7sus4 like Dom mentioned, or even a D locrian (a pronounced bass note D is an easy way to make that happen). But if it functions like an A? with a different "color/flavor" there's no need for that. If it's an informal situation I'd call it a "lydian triad" and try to make these names more well established, because they are useful in communication. About the symbol, it's a logical way for me to call it A?sus(?4) and I would, but that use is also at your own risk, since I've never seen it written.
No the chord as currently described is not a sus chord in its current form as sus chords don't contain ♯4ths. An A♭ sus4 chord would be spelled "A♭ - D♭ - E♭". Some people would call it an A♭#4, but I don't recommend that since it is a non functioning chord and very not standard.
Depending on what you are doing and how it is voiced, I may look at this as a type of E♭ chord specifically an E♭Maj7sus4 which is quartal in nature (E♭ - A♭ - D). This is more of the standard sus chord minus the 5th which makes sense due to the quartal nature of the chord. It's also highly possible that this chord is set up as a passing chord and for that reason doesn't make sense in typical harmony which is why deriving a name is difficult.
If Ab is considered the root, there is no third of C present or even implied. Therefore, Ab and Eb merely comprise a diad of Ab5, not a triad of ANY sort. In music theory, "sus" replaces whilst "add"...adds. Since there is no third which D replaces, "sus" is out. As previously stated, there is no triad on which we may "add" D, but there IS a diad. Therefore, Ab5add#4 is the most correct answer without any additional contextual information, although if such pertinent data was provided then one could advocate for Absus#4 as another possible solution.
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