I should note that I did reference the studio recording (I found an official upload on YouTube) and confirmed that what I noticed in the BBC version is also true for that.
Here's the alto recorder phrase:
After figuring it out and playing it about a dozen times, I thought it sounded vaguely familiar. Eventually, I realized that it has some resemblance to "Westminster Quarters," which is often used as a clock chime. I think slightly different versions exist, but it's something like:
The first measure is played at a quarter past the hour, the first and second measures are played at half past, and so on. At the top of the hour, the whole thing is played, followed by a single note (D in this key) played as many times as what hour it is (once for one o'clock, twice for two o'clock, and so on).
The alto recorder phrase in "My Name Is Jack" contains all of the same notes (plus a G); like "Westminster Quarters," it's fairly conjunct aside from dropping down to A; and the first measure of the phrase in "My Name Is Jack" matches the second measure in "Westminster Quarters" exactly.
I'm not sure if this similarity is intentional or whether it has any significance, but the two tunes do have a certain resemblance.
