We demonstrate a generalized method for dual-comb interferometry that involves the use of two frequency combs with quasi-integer-ratio repetition rates. We use a 16.67 MHz comb to probe an 80-cm-long ring cavity and a 100 MHz comb to asynchronously sample its impulse response. The resulting signal can be seen as six time-multiplexed independent interferograms. We perform a deconvolution of the photodetector's impulse response to prevent any crosstalk between these multiplexed data sets. The measurement is then demultiplexed and corrected with referencing signals. We obtain a measurement with a spectral point spacing of 16.67 MHz and a spectral SNR of 55 dB by averaging 15,000 interferograms, corresponding to a measurement time of 500 s. Compared to conventional dual-comb spectroscopy, this generalized technique allows to either reduce the spectral point spacing or the acquisition time by changing the repetition rate of only one of the combs.