We present a novel method to measure precisely the relative spectral response of the fluorescence telescopes of the Pierre Auger Observatory. We used a portable light source based on a xenon flasher and a monochromator to measure the relative spectral efficiencies of eight telescopes in steps of 5 nm from 280 nm to 440 nm. Each point in a scan had approximately 2 nm FWHM out of the monochromator. Different sets of telescopes in the observatory have different optical components, and the eight telescopes measured represent two each of the four combinations of components represented in the observatory. We made an end-to-end measurement of the response from different combinations of optical components, and the monochromator setup allowed for more precise and complete measurements than our previous multi-wavelength calibrations. We find an overall uncertainty in the calibration of the spectral response of most of the telescopes of 1.5% for all wavelengths; the six oldest telescopes have larger overall uncertainties of about 2.2%. We also report changes in physics measurables due to the change in calibration, which are generally small.
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