Clever, Ed! Given the lamp sync rate is synced to the ring cadence, I imagine you end up with this (an opportunity for ASCII art, yay):

Code

                       <---- 2sec ---> <---------- 4sec -------------> <---- 2sec ---> <------ 4sec..
                       _______________                                 _______________
RING CADENCE:  _______|    RING #1    |_______________________________|    RING #2    |_______________...
                       ___     ___     ___     ___     ___     ___     ___     ___     ___     ___     
RING FLASH:    _______| 1 |___| 2 |___| 3 |___| 4 |___| 5 |___| 6 |___| 7 |___| 8 |___| 9 |___|10 |___...
                       ___     ___                                     ___     ___
ED'S OUTPUT:   _______|   |___|   |___________________________________|   |___|   |__________________...
                       RING    RING                                    RING    RING

Hope I got it right -- Nice trick! And the 10VAC -> (Bridge Rect + 25V/1kuF cap) would make a nice clean DC for a relay coil.