Every tone means something unique to every tone. The Major Scale pattern creates the numbering system for all tones from the given root [1-7].
When we are using this system to create tone group [keys], this is called derivative. Derivative means that we are deriving the tone group from the Major scale pattern [221-2221]. This process 'creates' [221-2221 is a learning tool & has no inherent meaning in itself] what tones are in the diatonic family of the given root. Diatonic means "across the tones of a key". Diatonic tones are the tones that are in the Major key; the tones that are selected by this process.
The Major scale pattern gives specific tones, specific numbers (1-7 in traditional music theory naming). Again, in itself, the Major Scale Pattern [221-2221] doesn't really mean anything. It is just a learning tool. The five tones that were eliminated by the 2's are not derivative tones, but parallel. These tones are called non-diatonic; they are tones not in the diatonic family.
Parallel tones are tones which names are created by comparing to the derived tones (1-7). We typically [most commonly] parallel to the Major. We can also parallel to minor.
Every 2 in the Major scale Pattern creates a 'gap', since tones were 'skipped' or 'eliminated' at those points.
Again, the Major Scale Pattern [221-2221] creates a numbering for the tones from the given Root [R]. Yet, there are 5 tones to the root that are still unaccounted for [named]. So, we can compare those tones to the derivative tones. This is called paralleling. Parallel = Compare.
Since D is C's 2 [only C's 2], the tone that sits in the first gap is called D-flat. [flats lower tones one half-step]. We've compared a non-diatonic tone to a diatonic tone. In a key, the 2, 4, & 6 tones are the same tones as the 9, 11, & 13 respectively. The 2 = 9, & therefore, the flat 2 is also the flat 9. The sharp 2 = sharp 9, etc.
To clarify let's use an example. In C, D is the 2 [1-2]. If we keep going, beyond the octave [the 8th tone of the scale - the C], we find D again, yet this time in position 9 [1-2-3-4-5-6-7-8-9 = C-D-E-F-G-A-B-C-D]. Same thing applies to the 11 & 13…
2 = 9: 1-2-3-4-5-6-7-8-9 = C-D-E-F-G-A-B-C-D
4 = 11: 1-2-3-4-5-6-7-8-9-10-11 = C-D-E-F-G-A-B-C-D-E-F
6 = 13: 1-2-3-4-5-6-7-8-9-10-11-12-13 = C-D-E-F-G-A-B-C-D-E-F-G-A
The top row of numbers are the half steps [Numera].
The second row are the derivative and parallel names in the traditional music theory system.
C Major is the simplest inventory. Since all the derivative tones are naturals, the sharp and flat non-diatonic tones are the sharp and flat versions of the scale members. The flat names are shown above.
The D-flat could also be called C# [#1 - this is not common]; the E-flat could also be called D# [#2 or #9], the G-flat could also be named F# [#4], the A-flat could also be called G# [#5], and the B-flat could be also called A# [#6 - this is also not common].
The non-diatonic tones for C are F#/Gb Major pentatonic (D# minor/Eb minor pentatonic). We favor the tonal spellings of Eb minor/Gb Major pentatonic: Eb Gb Ab Bb Db for the non-diatonic tones.
We now know what all tones mean [are named] to C [we call this a tone inventory]. These tones hold a specific melodic & harmonic space to C. And, these tones only mean these to the tone C. In a different key, the tone D will be in a different position depending on the root. It is important to know the tone inventories for all keys. To help with this, please check out Tone Naming.