In this paper, we investigate the dependency between the spectral envelopes of speech in disjoint frequency bands, one covering the telephone bandwidth from 0.3 kHz to 3.4 kHz and one covering the frequencies from 3.7 kHz to 8 kHz. The spectral envelopes are jointly modeled with a Gaussian mixture model based on mel-frequency cepstral coefficients and the log-energy-ratio of the disjoint frequency bands. Using this model, we quantify the dependency between bands through their mutual information and the perceived entropy of the high frequency band. Our results indicate that the mutual information is only a small fraction of the perceived entropy of the high band. This suggests that speech bandwidth extension should not rely only on mutual information between narrow- and high-band spectra. Rather, such methods need to make use' of perceptual properties to ensure that the extended signal sounds pleasant.