Consider a simple haemonic motion of time period T . Calculate the time taken for the displacement to change value from half the amplitude to the amplitude.
Assume displacement = A sin wt
= A sin (2*pi*t/T)
max displacement is at t = T/4
Prior to that, 1/2 max displacement occurs when
1/2 = sin (2*pi*t/T)
(2*pi*t/T) = pi/6
t/T = 1/12
t = T/12
The time between half displacement and max displacement is T/4 - T/12, which is T/6