This is similar to one version of C9 but differs in having G not E top & bottom. I have always used this as a quick 7th but now wonder if it has a particular name. From bass E frets are 3, 3, 2, 3, 3, 3.
that is definitely a C9, more specifically C9/G, if you get rid of the sixth string G you'd have a regular C9.

And the shape you mentioned moves around like a barre chord better
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I would personally never play this inversion I don't think. Even on acoustic, especially being down near the nut, I find this would be too muddy sounding.

a x32333 though, or x3233x probably for a 9, and x323x3 for a 7, are chords that I will use often.

If you really wanted the G root, it depends on context exactly what I'd do, but that's too many close together bassish notes for my liking, but maybe more importantly, it's tough to play.

If you play a 3x2333 that's a similar sort of thing, still sounds think and beefy, and is a lot easier to play. That would be a Gm6.
Last edited by fingrpikingood at Dec 15, 2014,
Thanks Pastfarian96 & fingrpikingood.
I don't find it particularly difficult to play as I have said as I have used it for ages as a 7th substitute (with first finger on D-string, 2nd on E & A and 3rd on G, B and E ) and ,yes Pastafarian it moves up & down the fretboard very easily.