The differences between natural, harmonic and melodic minors

Answer

There are at least some differences in how they're used:
  • Natural minor is a kind of default.
  • In the common practice era a dominant chord usually contains the leading note, which is a semitone below the tonic, because that note has a strong tendency to resolve to the tonic. So, from the harmony point of view, the 7th degree of the natural minor scale should be raised in these cases, and you get the harmonic minor scale. Using a "flat" leading note makes the music sound modal, and was used both before and after the common practice era (probably also during it in folk music and such).
  • Raising only the 7th degree causes the scale to have an augmented second between the 6th and 7th degrees. This is a difficult interval to sing and also sounds like a jump, unlike the other intervals in the scale. So, from the melodic point of view, if you need the raised 7th and want to go up melodically, you also raise the 6th degree to make things more fluid.
  • Since the raised 7th degree is a leading note you usually go melodically upwards when using it. If you're going downwards you're probably not in a dominant harmony and thus you use the natural minor. Another reason for using the natural minor when going downwards is that if you take the upwards melodic minor and play that downwards it'll only start sounding like a minor after the fifth note or so.
In short, you start with the natural minor scale. You use the harmonic minor scale when you need a dominant chord, and modify it to be the melodic minor if you need the leading note in the melody. This all is, of course, simplified but should give the basic idea.

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