This station was originally authorized to construct at Brownfield, Texas in 2005. It was assigned the calls KCWV, which were changed to KMLU in 2006.
It never received a license-to-cover for operation in Texas. In 2007 it applied for permission to relocate to Kentucky, and in 2009 that permission was granted. ("major modification of construction permit". It doesn't happen very often, and it's VERY unusual to relocate several hundred miles, but it's not entirely unprecedented.)
So the K callsign moved to Kentucky with the station. The FCC apparently didn't require the station to change call letters when it moved across the Mississippi. It's a bit (but only a bit) more surprising they allowed the station to change the calls from KMLU to KYAI *after* it moved to Kentucky.
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For KTGG, the rumor has it the station requested WSAE. (to go with co-owned WSAE-FM) A Commission clerk saw "MI" as the station location and thought it meant Missouri. Since Missouri is west of the Mississippi, obviously WSAE is not a suitable call for a MI station & this station should be assigned the next K call available, KTGG. That is strictly a rumor.
I think KQV and KYW are mysteries. (KYW was initially located in Chicago but of course that's W territory too) KDKA was issued at a time when four-letter K calls were being assigned to *all* radio stations. That practice only lasted a few weeks, and the other two broadcast stations licensed during that period went out of business a few years later.
The K/W dividing line was not initially the Mississippi River. It was the western borders of North Dakota/South Dakota/Nebraska/Kansas/Oklahoma/Texas. (because the distinction was intended to be between maritime shore stations on the East/Gulf Coasts vs. those on the West Coast) A number W stations were licensed in those six states, as well as Minnesota, Iowa, Missouri, Arkansas, and Louisiana, before the dividing line was moved.
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Another long-distance move has brought another K call into the east just yesterday. KJWY, a full-power digital TV station, is now on the air from Philadelphia. (city of license is Wilmington, Delaware) KJWY moved from Jackson, Wyoming.