A theory is presented for the application of Hill's matrix method to the calculation of the reflection and transmission spectra of multitone holographic interference filters in which the permittivity is modulated by a sum of repeating functions of arbitrary period. Such filters are important because they may have two or more independent reflection bands. Guidelines are presented for accurately truncating the Hill matrix, and numerical methods are described for finding the exponential coefficient and the coefficients of the Floquet-Bloch waves within the filter. The latter calculation is performed by use of a computational technique known as inverse iteration. The Hill matrix for such problems is sparse, and thus, even though the matrix can be quite large, it may be efficiently stored and processed by a desktop computer. It is shown that the results of using Hill's matrix method are in close agreement with numerical calculations based on thin-film decomposition, a transfer-matrix technique. An important result of this research is the demonstration that Hill's matrix method may, in principle, be used to analyze any multiperiodic problem, so long as the periods are known to finite precision.