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The construction principles for a flexible X-ray spectrometer which can be equipped either with a single curved crystal or with two curved crystals are described. A few of the theoretical 4 + 32 possible working modes are selected for a closer investigation and examples of recorded spectra are given. It is shown in the general single-crystal case that for a wavelength close to a cut-off energy the narrow diffraction cone has an elliptic section, as predicted by the theory. The spectrometer is discussed in terms of intensity, resolution and dispersive power. A comparison with other types of spectrometer is made.