The prefix arch-, archi- “chief, principal; extreme, ultra; early, primitive,” derives from Latinized Greek arkh-, arkhi-, the combining form of arkhos “chief.” Usually, arch- is pronounced like “arch” (ɑrtʃ), and archi- sounds like “ark” (ɑrkɪ), although archangel (ɑrkeɪndʒəl) is a notable exception.
Is there an etymological or other reason for pronouncing the two prefixes differently? Did they perhaps enter English at different times? Likewise, is there any pattern to which nouns use arch- (e.g., archenemy, archfiend) versus archi- (archiepiscopal, architect)? Why is archangel an exception to the pronunciation rule?
Best Answer
An Etymological Dictionary of the English Language (1888) by Walter W. Skeat answers all your questions at once.
Another edition of this dictionary (revised in 1910; reprinted in 2005) adds a couple of comments to the explanation.