a modulated structure
Start with a plain, perfectly periodic crystal — the same as a row of identical fence posts, evenly spaced. Now add a slow, gentle wave on top: let each post lean a little, by an amount that rises and falls smoothly as you walk along, so post 1 leans right, post 5 stands straight, post 10 leans left, and so on over a long wavelength. The posts are still basically periodic, but a second, longer periodicity — the wave — has been laid over the first. A modulated structure is a crystal like this: a normal average lattice with a periodic modulation (a wave) superimposed on it.
The wave can modulate different things. In a displacive modulation the atoms are shifted from their ideal positions by a periodically varying amount (the leaning posts). In an occupational or compositional modulation the CHANCE that a site is occupied by atom A rather than atom B waves up and down along the crystal. Either way, the key number is the modulation wavelength compared with the base lattice spacing. If that ratio is a simple fraction like 3/1, the wave locks into the lattice and you just have a bigger ordinary unit cell — a commensurate (or 'lock-in') modulation, still an ordinary periodic crystal, sometimes called a superstructure. If the ratio is irrational, the wave never re-registers with the lattice and you have an incommensurately modulated structure, which is genuinely aperiodic. Diffraction reveals the modulation directly: strong main reflections from the average lattice, flanked by weaker satellite reflections whose spacing gives the modulation wave-vector.
Modulated structures matter because modulation is one of the main ways real crystals depart from the textbook ideal. They show up when a material sits near a phase transition (the modulation is often a frozen-in lattice wave, a soft mode, that has not yet locked in), in ordering alloys, in minerals, and in functional oxides. They are the gentlest form of aperiodic crystal — an ordinary crystal is still clearly visible underneath — and they are described with the same superspace machinery, usually needing just one extra dimension, (3+1), to become periodic again.
In gamma-Na2CO3 the sodium and carbonate positions ride a displacement wave. Diffraction shows the ordinary Bragg spots of the average cell, and around each one a pair of satellite spots at plus and minus the modulation wave-vector q. Measuring q fixes the wavelength of the shimmy; refining in (3+1)D superspace recovers exactly how each atom leans.
A modulated structure = average lattice + periodic wave; commensurate wave gives a superstructure, incommensurate wave gives an aperiodic crystal.
The satellite spots are usually far weaker than the main spots — miss them and you would refine only the average structure and never see the modulation. Whether the modulation is periodic-within-the-cell (commensurate) or irrational (incommensurate) is the crucial distinction.