One raised note, two consequences
Harmonic minor changes exactly one note of natural minor: the 7th degree goes up a semitone. The first consequence is the one it was invented for: a leading tone returns. The 7th now sits a half step below the octave and leans into it, so the scale arrives home with the same pull a major scale has. The second consequence is the price of the first. The 6th degree is still flat, and the gap from that flat 6th up to the raised 7th is now an augmented 2nd: three semitones, a step and a half, the widest move inside the scale.
That gap is the whole identity of harmonic minor. Everywhere else the scale moves by half and whole steps; right before the top it leaps. If you hear a minor scale climb stepwise and then jump near the end, you are hearing harmonic minor. The name records its job: Kostka and Payne note that the harmonies of a minor key mostly conform to this scale, the raised 7th supplying the leading tone the dominant chord needs.
On the staff
Listening for the gap
The C&M aural test for the minor scales is a two-step process, and harmonic minor is the one that ends it fastest. First, hear the minor 3rd near the bottom. That rules out major. Then listen to how the scale approaches the octave. Natural minor slides up smoothly. Harmonic minor does not: the flat-6th-to-raised-7th step lands as an audible gap, often described as exotic or "gapped," and then the leading tone resolves up by a tight half step. Gap, then snap. Nothing else in the scale set does that.
It is a sound with a long reach. The augmented 2nd is why harmonic minor turns up in flamenco, in Eastern European and Middle Eastern repertoire, and in neoclassical metal, anywhere a player wants minor with a sharp edge near the top. In functional classical writing it appears constantly, usually with the raised 7th sitting in the harmony rather than the melody, which is where the scale gets its name.
How Aubel handles it
Harmonic minor is the first scale to unlock after the starting pair, the third scale overall. By the time you reach it you have already drilled major against natural minor, so your ear is set on the minor 3rd. The new thing to catch is the augmented 2nd near the top, the gap that separates harmonic minor from the natural minor you already know. Rounds play the full scale up and down from a random root, so the cue you are learning is the gapped step itself, not its position in any one key.