The recorder in "A Dream within a Dream" (starting at ~1:03) is something like this:
In "(The System of) Doctor Tarr and Professor Fether," the recorder starts at ~3:29 and repeats throughout the rest of the song:
The keys are different, and some note values have been shortened in "(The System of) Doctor Tarr and Professor Fether," but all of the intervals are the same.
"A Dream within a Dream" and "(The System of) Doctor Tarr and Professor Fether" are the first and last songs, respectively, on the first side of the original vinyl album. Since this recorder part appears near the beginning of "A Dream within a Dream" and in the coda of "(The System of) Doctor Tarr and Professor Fether," it creates a sort of book-ending effect.