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Bach went to the trouble of writing all those rests in the bass notes -
if he wanted it to sustain he could have easily notated it so. In the
only recordings of the piece that I own, Eduardo Fernandez lets every
bass note ring until the harmony changes, while Kevin Gallagher usually
lets it ring, sometimes cuts it. Presumably they have good reasons for
doing so. What is one to do?
In Bream's 1992 recording of the Prelude, his damping approach is fairly
consistent. When the lower voice sounds alone, he lets it ring through. But
in measures where 2 voices are played simultaneously, he observes Bach's
rests pretty much as written.
In the Allegro he plays the rests as written. Bream comments in the liner
notes that "The problem of keeping the basses damped in the joyously flowing
Allegro is a lutenist's nightmare!"
Bream says there is doubt as to whether Bach's lute works are really written
for the lute. On the title page of the PFA autograph is written "Prelude per
la Liut o Cembal." The meaning of this is unclear, according to Bream. It
could mean "lute or harpsicord," or it could mean "lute-harpsicord," which
is a gut-strung keyboard instrument now extinct. (In my opinion, it's
perfectly clear, at least syntactically: the word "o" in Italian means
"or"). However, some lute scholars have concluded that PFA is probably a
lute-harpsicord work rather than a lute work.
Bach is known to have possessed several lautenwercks (lute-harpsichords). He
also designed and had built for him a thing called a lautenclavicymbel in
around 1740 (about the same time the PFA triptych was assembled). The
literal translation is "lute-clavier-harpsicord." Here's a good website
called "The baroque LUTE-HARPSICHORD: A Forgotten Instrument:"