  We extract the spectra of the strong low-frequency quasi-periodic oscillation
(QPO) and its harmonic during the rising phase of an outburst in the black-hole
binary XTE J1550-564. We compare these frequency resolved spectra to the
time-averaged spectrum and the spectrum of the rapid (<0.1s) variability. The
spectrum of the time averaged emission can be described by a disc, a Compton
upscattered tail, and its reflection. The QPO spectrum contains no detectable
disc, and the Compton spectrum is generally harder than in the time averaged
emission, and shows less reflection, making it very similar to the spectrum of
the rapid variability. The harmonic likewise contains no detectable disc
component, but has a Compton spectrum which is systematically softer than the
QPO, softer even than the Compton tail in the time averaged emission. We
interpret these results in the context of the Lense-Thirring model for the QPO,
where a precessing hot flow replaces the inner disc, and the harmonic is
produced by the angular dependence of Compton scattering within the hot flow.
We extend these models to include stratification of the hot flow, so that it is
softer (lower optical depth) at larger radii closer to the truncated disc, and
harder (higher optical depth) in the innermost parts of the flow where the
rapid variability is produced. The different optical depth with radius gives
rise to different angular dependence of the Comptonised emission, weighting the
fundamental to the inner parts of the hot flow, and the harmonic to the outer.
This is the first model which can explain both the spectrum of the QPO and its
harmonic in a self consistent geometry.
