Double time signature and strange bar notation in Bach

Question

So I have been studying Bach's prelude and fugue in d major of wtc II, where he uses two time signatures for the prelude. What does this mean? I did notice that some of the meqsures are written in a different time signature than others, but how should one play these alternating time signatures?

As a second question, in the e major prelude and fugue of book 2, the fugue in my edition contains a midbar vertical line, and even though it is alla breve it has 4 half notes per measure. What kind of time is this?

Answer 1

Bach is asking for a constant beat here, that is, where the measure is notated as cut time (e.g., ms. 2 & 4), the crotchet (quarter note) will match the dotted crotchet of the measures notated in 12/8. The giveaway here is when he writes dotted quaver (eighth note)-semiquaver (16th note) figures against triplet quaver figures: you'll note that the semiquaver lines up with the third triplet quaver in such cases. I've put a red box around the first instance of this. (You'll also note that the first beat of m. 3 uses crotchets in the outer voices against a quaver-4 semiquaver figure in the alto.)

Later in the Prelude, he makes the juxtaposition very clear by superposing a 12/8 melody over a variant of m. 2's motif:

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The E major fugue in Book 2 is alla breve where the beat is a semibreve (whole note). This fugue is in a very old-fashioned ricercar style, and Bach has used a very old-fashioned time signature for it (bearing in mind that units of beat have grown shorter over the centuries).

As an aside, this particular prelude-and-fugue combination is very interesting as Bach has paired the fugue with a very modern (for his time) prelude, and the combination works beautifully.

Answer 2

Patrx2's answer is a good explanation of the theory of the notation, but endless ink has been spilled in arguments about "how to play it" - and as with most music of Bach's time, there isn't a unique "right answer".

A straightforward interpretation is to take a fairly quick tempo (quarter notes at about 100 - 108) and play the rhythms "as written" following Patrx2's explanation.

On the other hand, you could take this at a slower tempo and "Frenchify" the rhythm. "French Overtures" were usually notated in dotted rhythms, but with the short notes written as longer durations than they were meant to be played. For example, at the start of the piece the 8th-note rest plus four 16-notes, would be played more like a quarter-note rest plus four 32nd-notes, though not necessarily in a "mathematically exact" way. In bar 5, the four written 16th notes F# E D C# in the right hand might be shortened to fit into the same time as the single 16th D in the left hand, though the left hand 16th would not be a mathematically exact 16th note - it might be a bit longer, to make room for the four right hand notes. This sort of thing is more effective on the harpsichord than on a modern piano, because of the more "explosive" attack from the plucked harpsichord strings - on a piano, the very short notes tend to get lost.

If you take the "French interpretation" to its logical conclusion (and perhaps beyond!) and treat the 4/4 8th notes as "notes inégales", they would not be played equally but have an agogic accent on the first note of each pair and therefore tend towards a triplet or swing rhythmic feel - and so we arrive at an interpretation of Bach's time double signature as meaning "the rhythmic feel of the whole piece is actually somewhere in between 4/4 and 12/8 - don't read any of the rhythmic notation too literally."

Some early editors took this to its logical (or perhaps illogical) conclusion and printed the whole movement in 12/8 time - for example https://imslp.org/wiki/Special:ImagefromIndex/62105. But since Bach was a practical performing musician as well as a composer, IMO if he had really wanted to write the piece in 12/8 he would have done just that.