harmonic minor, melodic minor--differences??

Natural minor = 1-2-b3-4-5-b6-b7-1
Harmonic minor = 1-2-b3-4-5-b6-7-1
Melodic minor ascending = 1-2-b3-4-5-6-7-1
Melodic minor descending = natural minor

From my understanding, you can use any one you want.

The big difference, besides the notes and intervals, is best seen in the perspective of history.

The 'natural' minor was used first. The problem with it, once you start to get into polyphony, is that it has no leading tone - that is, the seventh isn't a half step below the octave. That means a V-I cadence lacks the 'oomph' that it does in the major scale.

So the first solution was to raise the seventh, creating the harmonic minor scale.

The trouble with that scale is the big gap between the b6 and 7. Minor third intervals weren't that easy for singers, who wanted things more step-like. So composers experimented with raising the 6th too... but that meant the scale was the same as the major scale, except for the b3. It sounded a whole lot more major than composers were looking for when writing minor pieces.

The ultimate solution there was to use the raised 6th and 7th going up, but the natural minor going down. The result is called the melodic minor scale - readily singable, but more minor sounding.

Good answer, Noteboat.

My first violin teacher plumped up her theory with little historical asides like yours. It made stuff that could have been unbearably abstract and arbitrary a little more alive. Do you take a similar approach in your book?

Nope, sorry - it's dry like everyone else's. Great suggestion, though - I'll do that for the next editition.