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Spectrometry is virtually a synonym for Spectroscopy or Spectrography. For more information, see Spectroscopy.
It originated as a general term for the use of a Spectrometer, that measures Intensity as a function of Wavelength, with Spectroscope a more specific term for an instrument you "look through" (like a microscope or telescope) and Spectrograph for one that produces graphical output. Now virtually all data is collected digitally, easily displayed or printed graphically, so the term chosen for a specific instrument is insignificant.
Mass Spectrometry is an analogous study of the relative amounts of Particles based on their mass, i.e., the use of a Mass Spectrometer. In this usage, the convention is to use the word Spectrometry rather than the others.