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Let say I play the following:

LH: D G B

RH: F D

Overall would we say that I'm playing a G7 or some flavor of a Dm?

What about if the left hand is arpeggiated and the right hand is maintained? At first it sounds like a minor chord but then I feel it's more like a 7th chord. If we were to put a chord name on top of the bar which one should it be?

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If you are looking at the whole unit as one chord it's just a G7/D. While the bass note is important and helps shape the overall harmony, it doesn't make sense to look at it as a kind of D minor chord as the chord would then be adding an 11th and a 13th without a 7th or a 5th.

Playing it arppegiated will change the feel slightly, but does not change what you call it.

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It can only be known as G7. Spelled G B D F. Dm it won't be - there's no A, and a G crept in, along with a B. So at a push it could be Dm6add4(no5), because a lot of folk like to think of chords in root position, but let's just call it G7, second inversion. Or, as Dom says, G7/D.

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You can also look at it in terms of function and context: is the chord in question part of a cadence at the end of a phrase? Is it acting as a V7 of C (or CMaj7), perhaps? But it's still just a G7/D chord (i.e, V6/4 of C).

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