Minor key music uses the harmonic minor scale as the basis for its harmony and the melodic minor scale for the melody. If the music uses nothing but the natural minor scale it IS NOT in a minor key (that implies tonality, and tonal music requires the leading tone). It is modal Aeolian music.
I find this weird. Is this even true? If so, what does the quote mean by "harmony" and "melody"?
I would take this as a means to interpret what the author intended when writing the rest of the book. For example, if you play in a rock band and the band leader says a new piece you'll be learning is in D minor, there's no reason to assume that the piece is strictly functional harmony with no modal content. It could be in Dorian or Phrygian for all your band leader cares. It starts with a D minor chord and stays in that area for the whole song. It's all terminology. Changing what a piece of terminology means doesn't change how you expect the music to behave. So in this case, as reading through the rest of this text, just keep in mind that the author uses minor and modal terminology exclusively for specific contexts.
In common practice music, song in minor keys tended to use a major V chord rather than a minor v chord that the natural minor scale would harmonize to. V-i has a much more resolute feel than a v-i. That leading tone gives us an authentic cadence, which was the popular sound and is still one of the most powerful progressions in music. In A minor, that would be E-G#-B to A-C-E. The G# is altered from the natural minor harmony to give the E chord that leading tone. That's the harmonic portion. Only the seventh is raised for harmonic material. As for the melodic material, when melodies were written over this sort of progression, it was found that the interval between the minor sixth and the leading tone was too dissonant. It was, in fact, an augmented second. To correct this, the minor sixth was raised to a major sixth and the interval was much more pleasant. Typically, this melodic minor scale only has the sixth and seventh raised when ascending. Descending lines would use the natural minor.
So again, discrepancies in terminology are rampant in music because of mixed traditions, reusing terminology to mean different things, and lack of standardization. I don't know anyone who would lash out at you for calling a piece minor even though it uses modal melody and doesn't always contain a leading tone of the V chord. If the meaning gets across, that's what's important. This author knows what he means and makes it explicit, although he certainly seems to think that this meaning is more absolute than I'd agree with.
It is quite true and understandably very confusing wording at first.
The first thing you have to understand is how classic music theory defines tonality. It's very specific in that tonality is defined by the tonic - dominant relationship (V(7) - I or V(7) - i) which heavily depends on what is known as the leading tone. The leading tone is the note that is a semitone below the tonic and this gives us the major third in a V chord or a root in a viio.
Let's look at the A natural minor, A harmonic minor, and A melodic minor(ascending) scales.
A natural minor: A B C D E F G
A harmonic minor: A B C D E F #G
A melodic minor(ascending): A B C D E #F #G
The leading tone of any A related key is G#. In natural minor, this simply does not exist. We're not able to naturally build a dominant chord to perceive A as tonic as defined above. Using the harmonic minor scale however, we can now build two dominant chords using harmonic minor (E(7) and G#o(7)) which can be used to imply tonality as classic music theory defines it. Melodic minor also takes advantage of this while smoothing out the stepwise motion.
I'll link some more information about this in general.
I think what the author may want to tell you is that the natural minor form is specific to a melodic minor scale descending. It is not a scale in the strict sense of the word but rather what happens to the melodic minor scale when it descends. (The Sub Mediant and Leading Tone are lowered.)
