In this paper, we generalize and develop results of Queffélec allowing us to characterize the spectrum of an aperiodic
substitution. Specifically, we describe the Fourier coefficients of mutually singular measures of pure type giving rise to the maximal spectral type of the translation operator on
, without any assumptions on primitivity or height, and show singularity for aperiodic bijective commutative
substitutions. Moreover, we provide a simple algorithm to determine the spectrum of aperiodic
-substitutions, and use this to show singularity of Queffélec’s non-commutative bijective substitution, as well as the Table tiling, answering an open question of Solomyak. Finally, we show that every ergodic matrix of measures on a compact metric space can be diagonalized, which we use in the proof of the main result.