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I'm reading a book on music theory and it says

The harmonic minor scale is one of three traditional forms of minor scale. They are:

  • "pure minor scale" occurs as the 6th mode of a a major scale.
  • ascending melodic minor
  • harmonic minor

On this site and other literature uses the term "natural minor" scale. I think they mean the same thing but want to verify this.

Are they the same thing? Or is there a difference between the pure minor and natural minor scales?

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The book is using the term "pure minor scale" to refer to the natural minor scale. Yes; they are the same thing in this context, although "pure minor" is not a term I'm familiar with. It seems possible that the word "pure" is the result of an imperfect translation occurring somewhere along the line. But let's set that possibility aside and consider why the term "minor" might have been selected.

There are few reasons why the book might be calling natural minor the "pure" minor scale. For one, when we say "A min is the relative minor to C maj," we are referring to A natural minor (not A dorian minor, A melodic minor, or A harmonic minor). Secondly, when writing key signatures, there are generally two ways to interpret a single set of accidentals: as either a major key or a minor key. For example, if you see two flats on the staff, then you're working in either Bb major or G minor. In this context, the G minor tonality is built from G natural minor (not A dorian minor, A melodic minor, or A harmonic minor).

As a caveat, there are technically four minor scales, not three. The fourth minor scale that the book hasn't included is dorian minor, which is the second mode of the major scale.

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    I've never heard the term "pure minor scale". But if it's the mode based on the 6th step of a major scale, then it is indeed the same as the so-called "natural minor scale".
    – Scott Wallace
    Commented Aug 6, 2017 at 19:38
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    You guessed correctly about the context in which it was used which enhanced my understanding of the text . For anyone interested, the book is "The Jazz Language" by Dan Haerle (pg 21) which is discussing Modes of the Harmonic Minor scale.
    – PatS
    Commented Aug 6, 2017 at 22:14
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    If it's written by Dan Haerle, it's definitely not a translation issue. That was an intentional word choice. It's very interesting though that he didn't include dorian minor in the list, since it's perhaps the most commonly used minor scale in jazz. Maybe he's giving his interpretation of standard classical theory, and he thinks classical theory doesn't historically include dorian minor in that list of minor scales.
    – jdjazz
    Commented Aug 6, 2017 at 22:40
  • It's a pity there isn't a better term for a key signature than 'accidentals', 'cos that's what they're not. I guess the only term is actually 'key signature'.
    – Tim
    Commented Aug 7, 2017 at 5:46
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    Key signatures are not accidentals. In music, an accidental is a note of a pitch that is not a member of the scale or mode indicated by the most recently applied key signature. In musical notation, the sharp, flat, and natural symbols, among others, mark such notes—and those symbols are also called accidentals.
    – Richard Borders
    Commented Jan 21, 2019 at 1:37

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