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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-holebinary XTE J1550-564. We compare these frequency resolved spectra to thetime-averaged spectrum and the spectrum of the rapid (<0.1s) variability. Thespectrum of the time averaged emission can be described by a disc, a Comptonupscattered tail, and its reflection. The QPO spectrum contains no detectabledisc, and the Compton spectrum is generally harder than in the time averagedemission, and shows less reflection, making it very similar to the spectrum ofthe rapid variability. The harmonic likewise contains no detectable disccomponent, but has a Compton spectrum which is systematically softer than theQPO, softer even than the Compton tail in the time averaged emission. Weinterpret 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 isproduced 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 issofter (lower optical depth) at larger radii closer to the truncated disc, andharder (higher optical depth) in the innermost parts of the flow where therapid variability is produced. The different optical depth with radius givesrise to different angular dependence of the Comptonised emission, weighting thefundamental 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 itsharmonic in a self consistent geometry.

An imperfect double: probing the physical origin of the low-frequency QPO and its harmonic in black hole binaries