Goldberg variations - chord progression changes between variations

Question

The Wikipedia article on the Goldberg variations sets out the chord progression which all variations follow. In some bars, different options are separated by commas and it says there are different options taken by different variations.

What is meant by the comma notation here? In particular, what options are being indicated? For example, the first label is '3,5,6#'. Presumably one of the options is E major (E,G#,B), corresponding to '3,5' (or perhaps just to '3'?). What are the alternatives? In the case of 6#, is it supposed to be (E,G#,C#), (E,G#,B,C#), or perhaps something else? How many possibilities are being listed here?

Answer 1

TL;DR

By 3,5,6# (Kirkpatrick actually writes "3,5, or 6#" — the "or" making things much more clear) Kirkpatrick is saying that sometimes only the third above the bass is included (3), sometimes a complete root position chord appears (5), and sometimes a first inversion chord (6#).

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Kirkpatrick's explanation

Here is and excerpt of what Kirkpatrick has to say about the harmonic structure (see also the image below).

The fundamental bass is never state entirely in its most elemental form, as quoted here (Ex. 1), not even in the Aria....

Certain alterations to the fundamental bass are to be found. For example, chords of the sixth are interchanged with their root positions, and vice versa. It even happens that a six-four is substituted for a sixth-chord. The bass, the third, the sixth, or even the fifth is occasionally sharpened or flattened. In some places there is a certain interchange or ambiguous hesitation between the fifth and the sixth, with much use of the third alone against the bass, in order to leave this ambiguity free of limitation or definition.

Examples

Taking the third measure of Kirkpatrick's outline as an example, instances of each option can be found as follows.

3: Variation 1, m. 3

Variation 1 contains only the bass note (E) and the third above (G), leaving open whether it's a root position E minor chord or a first inversion C# diminished chord. Note that the C# occurring during beat 3 is not part of the bass-E chord. Rather, it's part of an A major chord leading to the D minor chord in the next measure.

5: Variation 3, m. 2

In variation 3, the first two beats of measure 2 comprises a root position E minor chord. The C# here is a passing tone and does not affect the harmony.

6#: The Aria, m. 3

Here is shown measure 3 of the Aria. It contains the chord E-G-C#, which is a first inversion C# diminished triad, the C# being a raised sixth above the bass E.

Answer 2

The numbers correspond to figured bass numerals of interval sizes (adjusted with accidentals as necessary) above the notated bass note. Note that interval sizes with no accidentals are assumed to take their notes from the home key only.

Note that the first appearance of the "3,5,6♯" figured bass numerals is above an E note in the bass. This means that you can play E-G, E-B, and E-C♯.

The true Goldberg variations according to your provided https://en.wikipedia.org/wiki/Goldberg_Variations#Form are more complicated than just picking among 3, 5, and 6♯. In fact, a good argument can be made that Variatio 5. a 1 ô vero 2 Clav. uses the entire 3/5/6♯ shebang with its use of E, G, B, and C♯:

Answer 3

These are the intervals played above the bass note. As we aren't able to notate them vertically like blockchords they are written with a comma. Usually the 3 and 5 isn't notated for every one it's clear they are played. 6 means first inversion. 5,6 means the dominant 7th chord first inversion.