For simple harmonic motion (that is, oscillatory motion for which the period is independent of amplitude), the restoring force must be proportional to the displacement. If gravity is the restoring force, as for a bead sliding frictionlessly on an upward-curved wire, then the angle of inclination of the wire must change as you move along it in such a way that the component of gravity acting along the wire, $g \sin\theta$, increases linearly with distance traveled along the wire. If you work out this shape, (see tautochrone) it is a cycloid, which starts out pretty flat, and curves upward more and more sharply. This is, then, the shape that a pendulum bob must follow in order to execute simple harmonic motion. In a simple pendulum, on the other hand, the bob follows a circular arc (constant curvature). Since the curvature does not increase with distance, as it must for constant period, the period must increase with amplitude.