Summary: We derive an exact sum rule satisfied by the dispersion relations of the commensurable case of Harper’s equation. We use this result to derive a lower bound for the total bandwidth of the spectrum and to provide a stronger analytical justification for a result due to D. J. Thouless [e.g.: Commun. Math. Phys. 127, No. 1, 187-193 (1990; Zbl 0692.34021)] concerning the total bandwidth when the commensurability is a high order rational.