Modern moiré theory is largely based on the Fourier approach. And yet, it seems that the Fourier theory by itself cannot answer all questions related to moiré effects. We present some visible phenomena in the superposition of periodic structures (such as line gratings) that are not captured by the Fourier approach. We discuss their significance, and provide possible explanations. In particular, we introduce the infinite module spanned by the frequencies of the original individual layers as a main tool in the moiré theory. We discuss its significance on the visible periodicities in the layer superposition-either real moiré effects, or pseudo-moiré (modulation) effects having no corresponding impulses in the Fourier spectral domain.