The great discovery of the Gothic architects was the practice of ribbed vaulting, wherein a series of stone ribs support the weight of a vaulted ceiling. The earlier, Romanesque style of architecture had employed barrel vaults, which are essentially solid stone tubes, shaped like the cross-section of a barrel. This design was structurally weak - while under construction, the vault had to be propped up with timbers to prevent it from falling apart. The completed vaults were prone to collapse, and because every stone in the structure was load-bearing, it was impossible to include any windows without risking the entire ceiling coming down.
In a ribbed vault, by contrast, the ribs bear the load of the entire ceiling, which is transferred to the ground through the piers on either side of the vault. This means that the ceiling itself can be made of lightweight stone, and that there is room between the ribs for stained-glass windows. The result is a style of architecture which is elegant, fills the finished building with light, and is in the long run structurally sound. All depends, of course, on the structural integrity of the ribs, whose arch shape is best suited to supporting a heavy load.
Anyone who has worked with choirs long enough to understand the basic principles of musical phrasing, of course, knows that the shape of the arch is crucial not only to the structure of Gothic buildings, but to the repertoire that was sung in them.