So I like Richard's Answer, and I think it's good, but I think it's worth adding as an addendum that his definition of "collection" actually disagrees with the author's. The author says :
The world scale, therefore, cannot properly be used to address a group or set of pitches in which no ordering is implied. That's where the word collection comes in. By using the word, collection, no step-wise ordering is implied. Instead, the word collection addresses a group of pitches arrayed around a single, tonal center (or tonic) pitch. Thus, a C-major collection means a collection of pitches in the major mode, arrayed around the pitch C.
Now, leaving aside that the author in the quest for terminological precision has used the very vague and difficult to define term "arrayed around" , the author clearly feels that "C major pitch collection" should denote the set of notes of the C major scale, with the tonic of C, but not necessarily arranged in the step-wise order of a "scale"
In this sense, it would be incorrect to describe the melody of Happy Birthday (below) a "using the C major scale" because there is no "scale proper" "true scale" (to coin new terms for an explicit, step-wise scale) in the melody. This would mean that it would only be correct to describe it as using the "C major collection".
T: Happy Birthday
|G3/4G1/4|A G c | B2 G3/4G1/4|A G d |c2 G3/4G1/4|
|g e c | B A f3/4f1/4|e c d | c2
According to the author's definitions, it would be incorrect to refer to Scarborough fair as using the "C major pitch collection" though, as it is centred around the "D dorian pitch collection"
T: Scarborough Fair
K: D dorian
|D2 D | A A A | E3/2 F1/2 E | D3 | z A c | d2 c | A B G | A3 |
| z2 d | d2 d | c2 A | A G F | (E1/2C1/2-C2) | z4 | D2 A | G2 F | E D C | (C1/2D1/2-D2)
However, Richard makes the following point:
In my experience, the distinction between "scale" and "collection" mainly becomes important when dealing with twentieth-century (and beyond) techniques like modes. We would never say that "the C-major scale is the same as the E-Phrygian scale," but we can say that "the C-major collection is the same as the E-Phrygian collection." It's a bit like the square and rectangle issue: a scale is a collection, but a collection isn't necessarily a scale.
This means that, according to Richard, it could be perfectly correct to describe Scarborough Fair as using the "C major pitch collection", since it uses only the notes found in that collection.
So in our quest for more precise terminology, we have ended up with a different ambiguity. We've resolved the ambiguity of the word scale meaning both:
1) a set of notes played sequentially in a stepwise motion
2) a restricted set of notes, with a tonic, used to construct a piece of tonal music
but introduced our own definitional ambiguity for "collection" of:
1) a restricted set of notes, with a tonic, used to construct a piece of tonal music
2) a well defined set of notes, in any order, in any musical context
And in addition to this new ambiguity, I don't think the original ambiguity for the word "scale" actually causes problems, because it's clear from context whether we're talking about an "actual" sequential scale being played in a piece of music, vs "the scale" as a set of notes used to construct music.
Much like the word "alphabet" might mean "the set of letters used to write a language" and it could also mean "ABCDEFGHI..." in order, the difference between "English uses the Latin Alphabet" vs. "He wrote out the English alphabet" is clear from context, as is:
"the flutes in this passage play rapid C major scales"
"this melody uses the C major scale"
If disambiguation is really required you could say you could substitute scales and scale respectively for "stepwise C major scales" or "the notes of the C major scale", but imo neither is required in either of my (relatively typical) examples.
In conclusion, to borrow the verbage of to the author:
Now, I know this might sound like terminological nit-picking, but, for these reasons, I'll keep using "scale" for both of its conventional definitions. Trust me on this ;)