Tower puzzle

CAN you help Graham Thomas find out what this old building in Annan once was. He said: “This strange old tower with a pyramid shaped roof is in the centre of Annan at Greencroft. It has one unglazed triangular window. If you look at the stonework you can see that it once had a similar one lower down the tower. “I’m trying to work out what purpose it served when new. I know that pigeon racing used to be very popular in the town, with birds being sent to provincial towns in France and released en masse to race home. I wonder whether it could have been a pigeon loft?”