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I have a chord with the notes G C D Eb. I wonder how I write it down in a jazz notation.
First, something like this comes to mind: G-6(sus4). But 2 things confuse me about this. First, I've never seen a suspended chord with a minor/major indication because it doesn't matter for a triad, which is usually the example. Second, there is a C-6 chord in "My Funny Valentine", and it's played with a major 6, not with a minor, or at least, that's what I was told.
With those 4 notes the logical choice is Gsus4(b6). There are probably several other options I can think of but this symbol will give you exactly those 4 notes. It will also be spelled as a root position chord.
G-6(sus4) is incorrect because a sus4 chord is neither major or minor. The “-6” implies a Bb and an E (M6) in the chord the way you wrote it.
There are other options because of the 3rd intervals in the chord, that is making it an inversion of a Cm or Eb but I believe you put the G first because you hear that as the root.
Cm add9 works o.k., or if you need a purist's name, Cm add9/G. Looking at the notes involved, it would appear to be a C rooted chord, hence '/G' to voice it better..
You really, really should put this into the actual musical context to show how you are using it. There is no voicing or octave information. Nothing about how you might approach and depart the various tones involved.
Two voicing that to my ears suggest fairly strongly alternate identities for the base triad are...
But you can also get a stack of perfect fourths from those pitches which could be going in a 'quartal' direction or more conventionally suggest a suspension in a dominant chord...
If you meant to give the pitches as close voicing, I still don't know what octave you are in...
...both of which sound like tone clusters to me in isolation. In chord symbol form, I suppose Cm(add2)/G best conveys the close position, cluster voicing.
These aren't academic theory, hair splitting questions. It's a question of speaking a musical language, and when you write it down to make yourself sensible to other musicians. If your musical intention were the D7♭9(sus4) voicing, where musically you are expecting the sound of a dominant ninth chord in G minor, with a suspension (a mouthful to say, but pretty straight forward harmonic idea), but you wrote down Gm(sus4/sus♭6), it would be misleading and confusing.
One additional thought about a sixth over a chord root, especially of a minor tonic. A standard minor key signature uses a minor sixth above the tonic. But it's pretty unusual to see something like, in for example C minor, a tonic Cm(♭6). The main reason being that chord aurally sound much more sensibly like A♭maj7/C. The more common jazz chord is Cm6 where the sixth is major. As you pointed out, that major sixth is not diatonic to a minor key signature. But, that chromaticism is part of the language of jazz.
That chord does appear in the song My Funny Valentine, but keep in mind that particular chord is formed by a descending chromatic line above the main bass. The chords being Cm Cm(maj7) Cm7 Cm6 where above a steady bass on C you have a descending line C B B♭ A, after that it goes to a A♭/C chord and that descending chromatic line resolves to the Ab of the A♭/C chord. You can think of the whole passage of Cm Cm(maj7) Cm7 Cm6 as just Cm for four bar with a bit of decorative chromatic counter point on top that eventually resolves to a consonant, in key signature A♭ in the A♭/C chord.
I don't think it has much bearing on your main question of G C D E♭, but you did bring up the diatonic/chromatic aspect of the sixth above a minor chord root, sixth degree of the minor scale, and the song My Funny Valentine, so I thought I try sharing some detail about it.
