Notating The Chromatic Scale

The so-called "chromatic scale" consists entirely of a 12-note array of consecutive semitones; it contains each and every black and white note on the keyboard between its starting note and its octave.
When we find different organist/composers notating this scale in their scores in arbitrary ways (albeit consistently, the actual pitches being exactly the same, and readable enough) it begs the question about what method of notation might be best to adopt and how one would arrive at that.
In theory the chromatic scale, just like the minor scale, exists in 2 different forms -- harmonic and melodic -- both of which proceed by semitones.
The harmonic form is so named because its semitones can be harmonized within the normal diatonic (step-wise) scale structure; the melodic form, on the other hand, was invented because it has fewer accidentals and is easier to read.
In its harmonic form, scale degree 1 (the tonic) and scale degree 5 (the dominant) are written first and only once on the staff; all other (10) chromatic semitones are written twice, i.e. in pairs between the tonic and dominant notes.
The melodic form is used ascending only; in descending the harmonic form is used simply because there is no descending form of the melodic chromatic scale.
In the melodic form, scale degree 3 (the mediant) and scale degree 7 (the leading note) are written first on the staff and only once, and all the other notes twice, paired where possible.
When the chromatic scale over one octave is written ascending or descending, the ending note at the octave needs to be the same letter and accidental as its starting note.
Major scales and both forms of the minor scale are called "diatonic" because each note has a different letter name; neither form of the chromatic scale can be diatonic, however, as letter names are repeated.
In actual practice composers approach writing an extended chromatic scale in their scores basically in 2 different ways:
1. One way is by using accidentals on a blank staff implying a C Major/a minor signature -- in this form no letter name should occur more than twice in a row.
For example, one would not use Gb, G natural, and G# -- instead Gb, G natural, and Ab would be used.
If the starting note is a natural or sharp, then sharps are used ascending and flats descending, the only caveat being that the first and final note must be notated the same (photo).
In other words, if the first note is a sharp, then the last note must be sharp as well -- if the first note is a flat, then the last note must be flat.
If the starting note happens to be a flat, then flats and naturals are used until we can switch without breaking the "two-letter-rule."
2. The other way, which is far more common, involves notation which is subject to the sharps and flats in all the other key signatures generated by the Circle -- the method is to write the chromatic scale within a sharp or flat key signature and add in notes with accidentals, bearing in mind that the interval between the mediant (scale degree 3) and subdominant (scale degree 4) of the major scale, as well as the interval between the leading note (scale degree 7) and the tonic (scale degree 8), is already one semitone.
This provides the organist/composer with a guideline to follow when notating the chromatic scale which can be applied for any key signature beginning on any note.
In an Eb Major signature (3 flats), for example, the melodic chromatic scale would be applied in the score ascending and would observe each note flattened by the signature (Bb, Eb, Ab), use the mediant note (G) and leading note (D) just once, use the other notes no more than twice, pair these other notes by letter where possible, sharpen the naturals where needed, and would be notated this way:
Eb, E, F, F#, G, Ab, A, Bb, B, C, C#, D, Eb
Descending the harmonic form would be applied and would still observe the notes flattened by the signature (Bb, Eb, Ab), use the dominant note (Bb) just once, use the other notes no more than twice, pair these other notes by letter where possible, flatten the naturals where needed, and would be notated this way:
Eb, D, Db, C, Cb, Bb, A, Ab, G, Gb, F, Fb, Eb.
The coda of Processional March in Eb Major Op. 3 for hands only is an illustration of this [See blog, Bio/Free Stuff, !0 Pieces -- FREE].