A general condition for band gaps is given in the form of a (generalized) continued fraction. By use of this condition and of the convergence theorem of continued fractions known as Worpitzky's theorem, the critical mass ratio is rederied for `special frequencies' at which vibrational frequency spectra of isotopically disordered diatomic linear chain vanish independently of the concentration and arrangement of atoms. The notion of special frequencies is also applied to the electronic system in the Hückel approximation. The intergrated spectrum at each special frequency can be easily derived by virtue of the property of special frequencies. The Saxon-Hutner theorem is extended to the two- or three-dimensional Rosenstock-Newell model with constant non-central force constant. By virtur of this, the possibility of the existence of band gaps in the presence of a certain kind of short range order is pointed out in any dimensional aperiodic system.