In the wall frame of a timber building a bay is the space between two sets of principal posts, so a three-bay house has two end walls and two pairs of intermediate posts. Local building practices meant that ‘bay’ came to be considered as a standard width although it differed from region to region.
In Old English ‘beam’ was a tree and this meaning survives in compound names such as ‘hornbeam’. It came to refer to large pieces of squared timber, as used in house-building but was not very common in Yorkshire.
The noun ‘bilder’ was the name given to a wooden implement used for breaking up clods of earth (EDD) and Watson, the Halifax historian, said of ‘bildering’ that it was ‘Levelling the ground and breaking the clods of earth’. He considered it to be from ‘billing’; that is using a ‘bill’.