What is the origin of the phrase "take it from the top?"

"Take it from the top" is often used in musical rehearsals to mean "start at the beginning." Where did this phrase come from?

Here's what the Oxford English Dictionary says:

Music colloq. (orig. Jazz). from the top: from the beginning of a piece of music, typically for a second or subsequent time; also in extended use. Freq. in to take it from the top : to start (over) at the beginning.

1940 Gramophone Nov. 140/3 (title of song) Take it from the top.

However, Robert S. Gold's A Jazz Lexicon (1964, read online) says it's earlier by a few years:

top (down), from the, [by analogy with reading sheet music; current since c. 1930] See 1936 quot.; also, by extension, from the beginning of anything (to the end). — 1936 Metronome, Feb., p. 21. from the top down: playing an orchestration right through. —1956 Eddie Condon's Treasury of Jazz, p. 219. "Let's do this one more time from the top, gentlemen."

This also points out it can mean not only to start at the beginning, but also to keep going until you get to the end; to initiate a full run-through.

I think they're referring to the physical top of the song's or score's first page. I've always taken this literally, which seems to work.