The pendulum law is a small-angle approximation. It is exact only if the swing is tiny. In a real pendulum, the push that brings the bob back to the middle gets weaker, relative to the distance, as the swing gets wider, so a wide swing takes slightly longer than a narrow one.

If the swing reaches an angle θ₀ on each side of vertical (the amplitude, in radians), the true period is longer than the small-angle value T₀ by about:

T ≈ T₀ × (1 + θ₀² / 16)

This means a clock whose pendulum swings wider will run slower. The effect is called the circular error, because the bob moves along a circle (an arc) and not along a straight line. A perfectly isochronous pendulum, one whose period is the same at every amplitude, would have no circular error. Real ones are only nearly so, and only for small swings.