Hey Presto

You find "Hey Presto" all over the shop, even here where it's used, for example in explaining the origins of "On the QT", but where does "Hey Presto" come from?

I assume it is the same 'presto' as in the musical term meaning 'quickly', which is Italian, and in turn from Late Latin praestus (adj) quick, ready to hand, and Latin praestō (adv) present.

DFG