This is to explain the essential theory behind Harmonic Minor in a very brief way. For you to understand this it is necessary to know what it means to harmonize a scale. This you should have worked through with the Ionian scale already.
In the Ionian scale system, on the VI mode we have “Aeolian”, which is the parallel minor key to the root major key. For C Major this looks like that:
with the chords: Cmaj7 – Dm7 – Em7 – Fmaj7 – G7 – Am7 – Bm7b5
Now we switch the root note to the VI, and we get the Aeolian scale:
with the same chords, only the relations to the I are changing:
Now comes the point: If we want to have a “proper” tonic function in minor· we want to have a dominant chord on the V. The particular tension on the V7 chord (with 1 3 5 b7), and basically the tritone-tension between the 3 and the b7 in this chord, defines the function as a dominant-7 chord.
But so far we have a minor7 chord on the V of Aeolian. Therefore we just change one note, in this key it is the g becoming a g#, and we do have a E7 (dominant 7) on the V, and as a result we do have “A Harmonic Minor”:
That’s it, and from here on we can create a whole new scale system when harmonizing this scale as well.
Here is a pdf for all the modes of “F Harmonic Minor”. Once the method is understood, you can write it down for all 12 keys in the western music system.