Based on an initial expectation from laboratory measurements or instrument simulations, photometric passbands are usually subject to refinements. These refinements use photometric observations of astronomical sources with known spectral energy distribution. This work investigates the methods for and limitations in determining passbands from photometric observations. A simple general formalism for passband determinations from photometric measurements is derived. The results are applied to the passbands of HIPPARCOS, Tycho, and Gaia DR1. The problem of passband determination is formulated in a basic functional analytic framework. For the solution of the resulting equations, functional principal component analysis is applied. We find that, given a set of calibration sources, the passband can be described with respect to the set of calibration sources as the sum of two functions, one which is uniquely determined by the set of calibration sources, and one which is entirely unconstrained. The constrained components for the HIPPARCOS, Tycho, and Gaia DR1 passbands are determined, and the unconstrained components are estimated.