Draw the orbit in phase space, then play the piano keys...
...or draw your waveform directly!
(Shows the level of individual overtones in the above waveform.)
(Shows frequency content of audio output over time.)
To second order, we have
$$ x_{i+1} = x_i + \delta t_i \dot x_i +\frac{(\delta t_i)^2}{2}\ddot x_i+\dots, $$yielding our approximation for time intervals \(\delta t\)
$$ \delta t_i^{(3)} = 6\left[ \frac{x_{i+1}-x_i}{4\dot x_{i+1}+ \dot x_i + \dot x_{i-1}} \right]. $$