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I am currently analyzing a bit of music for homework which consists of four voices (string quartet).

It is in the key of G Major, but the chord that it ends on, is a Ab on the cello, B on the Viola, and G on both violins.

So that chord is Ab B G, does anyone know how I would figure this in a roman numeral analysis?

Here is the score

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The Ab in the cello part in the final chord of the music you posted is almost certainly a typo. The entirety of the piece aside from that is written with typical common practice harmonies that you would expect to see in a theory class.

The only context I would expect to see the music played as written would be a mistake on the part of the cello player or typesetter, or a bizarre piece of performance art that exists to make the audience cringe when the cello plays the last note. IIbmMaj7 chords do not exist in Theory 101, and when you see that combination of notes in a theory class, you're not going to be calling it a IIbmMaj7 chord anyway (you'll be calling it a 0-1-4 pitch class set).

It's supposed to be a G natural in the cello part, making the final chord a perfect authentic cadence on the I chord.

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When identifying a chord, I normally follow these four steps. Here, I have them written out with the specific steps for the chord listed in the question.

  1. First, when identifying the roman numerals, we must identify the notes that appear naturally in the key. Here, the key of G Major naturally contains the notes, G A B C D E F#.

  2. Identify the notes in the chord. Here, we have Ab B G.

  3. Identify where these notes fall into the the natural ascending scale of the key. A is II. So, Ab would be IIb. B is III and G is I.

  4. Finally, we find the name of the chord. So far, our chord is IIb III I. When we see a IIb with a I in the same chord, it usually means it is some sort of a seventh chord. Here, the III would be the (minor) third, making it a minor chord with a major seventh added. This forms an IIbm(maj7) (Ab minor with a major seventh added) chord in root position. The fifth (VIb/Eb) has been dropped by the composer because the ear will naturally "hear" it whether it is played or not.

So, our end result is IIbm(maj7)

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