When you have a song with all white notes except for G#, then what you have is a song in the belly-dance minor. (I listened to the video, and that's right.) I believe piano teachers call it the harmonic minor.

To start from the beginning:

1. Any major scale goes do re mi fa so la ti do.

2. Pick any major scale, start playing it, and observe what its 6th note is.

3. Now play a scale that starts on that 6th note. Keep the sharps and flats the same as before, in Step 1. That's the melodic (I'm pretty sure) minor scale. At my house it's called the ordinary minor.

4. Do step 3 again, only this time, when it's time to play the seventh note, sharp it by one half-step. That's the belly-dance minor.

A B C D E F G# A

Despite its name, the belly-dance minor is capable of being mournful, reverent or saucy, depending.
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A word about the official terms: I have always thought the terms 'melodic minor' and 'harmonic minor' very silly. I can compose harmony in the melodic or a melody in the harmonic. The names convey nothing. So I call them 'ordinary' and 'belly-dance.'