For Grade 3 Piano it says that I can choose between harmonic or melodic minor scales.
Could you please help me understand what they are and the difference?
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What you are probably talking about is harmonic MINOR and melodic MINOR scales. These are minor scales derived from the natural minor scale W-H-W-W-H-W-W (being W: a whole step and H: half step) but with one or two ascendings tones, the 7th, and the 6th, respectively. That is the harmonic would be W-H-W-W-H-(W+H)-H. Note that between 6th and 7th degrees there is a space of augmented second. Therefore this is "fixed" or smoothed with the melodic scale ascending the 6th degree also a half step so that the melodies constructed towards the tonic -generaly above the V chord- don't get a succession of augmented second interval (which causes dissonance). The melodic minor also uses flat 7th and flat 6th (specially when not above the V chord) when the melody is descending. Then we could say the melodic minor has 9 tonal notes.
So the seventh degree is ascended in the first place, in order to gain more tension towards the tonic, and the 6th is added so that when you ascend that 7th you don't get a augmented second. Now you should hear that maj7 in a minor scale and feel that really strong tension created and you will get the idea why that note is ascended in the first place. The Melodic minor then would be W-H-W-W-W-W-H when ascending and W-H-W-W-H-W-W when descending.
Written in the scale of A minor you would get:
A Natural minor: A-B-C-D-E-F-G-A
A Harmonic minor: A-B-C-D-E-F-#G-A
A Melodic minor: A-B-C-D-E-#F-#G-A (asc) and A-B-C-D-E-F-G-A (desc)