Lighter wrote: But why Sir "Jasper"? That's a somewhat unusual name, isn't it?

Maybe he was hard as a rock? (Growf. The things I think of when dealing with bawdy songs.) It's not a common name, but it's not extremely unusual; I believe I knew a man named "Jasper" half a century ago, and there is a radio host with a cat named "Jasper."

The song is in the DT -- file SIJASPR.

Roud mixes it with "Pink Pajamas" (both #10311), but I certainly wouldn't.

Averill's Camp Songs, Folk Songs cites it three different times, so it likely was popular in camps although, of course, no camp songbook would include the thing.

I have not found any early references to it, but there are some interesting notes which I derive in part from Averill:

The trick here, of course, is that each time through, one more word is left out (perhaps with muffled sounds replacing it), until the final sounds are almost orgasmic.
It really should end with a "Tee hee," at least if anyone knows Chaucer's "Miller's Tale."
I would love to know what camps allowed this thing!
Averill calls this sort of thing, where words are left out of a song, a "decremental song." All I can say is, in terms of cleverness, this stands head and shoulders above the rest, even if it is not usable in polite company.
"John Brown's Body" seems to be a popular tune for decemental songs, being used for both this and "John Brown's Flivver." - RBW