  In the post-Newtonian (PN) regime, the timescale on which the spins of binary
black holes precess is much shorter than the radiation-reaction timescale on
which the black holes inspiral to smaller separations. On the precession
timescale, the angle between the total and orbital angular momenta oscillates
with nutation period $\tau$, during which the orbital angular momentum
precesses about the total angular momentum by an angle $\alpha$. This defines
two distinct frequencies that vary on the radiation-reaction timescale: the
nutation frequency $\omega \equiv 2\pi/\tau$ and the precession frequency
$\Omega \equiv \alpha/\tau$. We use analytic solutions for generic spin
precession at 2PN order to derive Fourier series for the total and orbital
angular momenta in which each term is a sinusoid with frequency $\Omega -
n\omega$ for integer $n$. As black holes inspiral, they can pass through
nutational resonances ($\Omega = n\omega$) at which the total angular momentum
tilts. We derive an approximate expression for this tilt angle and show that it
is usually less than $10^{-3}$ radians for nutational resonances at binary
separations $r > 10M$. The large tilts occurring during transitional precession
(near zero total angular momentum) are a consequence of such states being
approximate $n=0$ nutational resonances. Our new Fourier series for the total
and orbital angular momenta converge rapidly with $n$ providing an intuitive
and computationally efficient approach to understanding generic precession that
may facilitate future calculations of gravitational waveforms in the PN regime.
