Why do we say "archenemy" differently from "archangel" and "architecture"?

Solution 1:

I am not a native speaker, but I see a major difference between arch- in archenemy or in archaeology and even another one to archipelago, which would explain the different pronunciations.

In the first case it is used as a prefix. Enemy is still a word by itself, as bishop or diocese are. The arch- prefix is used to emphasize the relevance / importance / significance of this special enemy / diocese / bishop. Here arch- derives from the Greek archi- or arkhi- (main, chief) as you mentioned before.

This is also correct for archipelago (archi meaning main), but here it is a word that can only be used as a whole. Pelago has no meaning in English. Archipelago derives from Italian (where it comes from Greek), so one cannot compare it with archbishop etc. It has not been combined from already English words.

In archaeology arch- is no prefix. Instead archaeo- could be counted as one. The word derives from the Greek words archaios (old, archaic) and logos (science).

The difference becomes clearly visible when translating to German:

Archbishop - Erzbischof

Archenemy - Erzfeind

Archdiocese - Erzdiözese

Archipelago - Archipel

Archaeology - Archäologie

As a prefix to the given words Erz- has the same meaning as arch- and like archipelago the German word Archipel derives from the Italian word. Archaeology and Archäologie are just the same.

Solution 2:

Further to John Lawler's comment to the OP: Here is one factor which could explain how come the "arch-" prefixes of archenemy and archangel are pronounced differently. Archangel is derived from the Greek archangelos, that is, the compounding was done in Greek. By contrast, archenemy was formed in English by combining the English prefix arch- and the word enemy. True, that prefix was derived from Greek, but its pronunciation has been Anglicised. The same applies to Daniel's examples archbishop and archdiocese, and also to archdeacon, archduke and archlute.