Meaning of numbers under bass line in BWV 51 mvt.3 (maybe chords?)

Question

Natalie Dessay performs Bach's very beautiful Aria "Höchster, mache deine Güte" (A minor) in these two videos

(only the sound and the score)

(with video of performance)

The score is available at IMSLP: http://imslp.org/wiki/Jauchzet_Gott_in_allen_Landen,_BWV_51_(Bach,_Johann_Sebastian)

The score is originally for cello (basso continuo) and voice: however, the performers also use an organ, which is not listed among the instruments, with particular chords: Am, Dm7, G7, C, F, Bdim, E7, Am, ... (B stands for B natural)

I have a feeling that the numbers below the cello part (7, 7, 7, 7#) have some particular chord meaning. In the sequel, other numbers appear:

Clearly those numbers don't correspond to fingering because the number of fingers is limited by 5. Those number also don't correspond to strings for the same reason. The number 7 for the sept-chords could be a bit confusing because the 5-th number seven in the beginning (B-C-D) sounds like a diminished chord Bdim, but then the sharp (E-D-E) is logical because we use the dominant sept-chord with G#. Anyway, I can't extend this logic because I don't understand the meaning of "6-4-3", "6", "6-5", just "#", "9" because I could try to interpret them in terms of some jazz chords but this doesn't fit into Bach's barocco style, and also doesn't fit to what I hear on the recordings.

Do these numbers have particular meaning? If they correspond to chords, is there some particular algorithm to convert these numbers into chords? Or maybe this organ part is just a custom arrangement of the performers?

Answer 1

Those are bass figures indicating a harmony to use in connection with the bass note. It is a rather elaborate system that keyboard players at Bach's time were expected to be able to improvise on. Nowadays scores written in figured bass are usually instantiated by the publisher.

If the harmonies sound jazzy to you, that's partly because they are. For one thing, the bass line often leaves the root note of the harmony and thus the spelling of the harmony has to adapt. For another, chord changes in figures bass tend to be a whole lot more often than nowadays. And then lots of the progressions used at that time were of modal nature where our mainstream music often is just major/minor these days and modal harmonizations are mostly known from Jazz.

I am somewhat bemused because you state that Jazz chords don't fit into "Bach's barocco style": Bach is not Telemann. Part of the reason he was only moderately popular in his own time were his radical harmonizations that pervade his later organ works and fugues in various instrumentalizations.

Answer 2

A little score example showing some similarity to your first snippet:

the figured bass is taken from: Georg Philipp Telemann "Singe-, Spiel- und Generalbaßübungen¹" (Hamburg, 1733/34). You find the original here on IMSLP (Page: 22; Title: "Die durstige Natur"²)

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Explanation:

• 1st Chord: you add a third (G♯ => B); a fifth (G♯ => D♯) (we don't use the actual flattened fifth from the VIIth degree of Amajor here) and the 7th (G♯ => F♯).
• 2ond Chord: we use root (C♯); add third (C♯ => E) and the 7th (C♯ => B).
• 3rd Chord: fifth and 7th of F♯ can be used from the previous chord (C♯ and E); A is the added third.
• 4rth Chord: the ♯ underneath the last 7 indicates to use a major third (D♯ instead of D); B is the root and A the 7th.

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¹ Generalbass (german) = figured bass
² "Die durstige Natur" (german) = "the thirsty nature"