The authors calculate exact upper and lower bounds for the Helmholtz free energy of a 1D tight-binding disordered alloy as a function of temperature. They show that the average of the upper and lower bounds, calculated using very few (of the order of five) exact moments of the density of states, is an excellent approximation to the free energy of the system. They also use three different approximations for the density of states of the disordered alloy and show that they all give quite similar and precise values for the free energy. Thus, if only the free energy of a disordered system is required, the average of upper and lower bounds seems to be the most convenient procedure to use. If other properties are calculated as well, then obviously some approximation to the density of states will be needed, which will certainly be sufficient for the calculation of the free energy.