Who's Afraid Of The Big Bad Scale?

Everybody knows
C-2-D-2-E-1-F-2-G-2-A-2-B-1-C
          with its strange mixture of tones (indicated "2", for 2 semitones) and
          semitones (indicated "1")
     which we call the "scale" and treat with reverence,
     considering it the fundamental corner stone of Music.
          Once again we speak in terms of equi-tempered tuning here.

Why?
The curious observer cannot help but ask:
     1. why these unequal sizes;
          is it possible to link these 7 notes with a constant distance?
     2. why are the semitones placed between E-F and B-C,
          rather than elsewhere?

Constant distance
     Let's try the first question and see if we can find a constant distance.
     Again starting on the note C, let's take each second note:
C-4-E-3-G-4-B-3-D-3-F-4-A-3-C
     No luck. We still have a mixture of
          3 semitones (minor third) and
          4 semitones (major third).
               We recognize the major and minor Triads here.
          Let's not give up.
     Starting this time on the note B, let's take each third note:
B-5-E-5-A-5-D-5-G-5-C-5-F-(6)-(B)
          The B must be at the beginning because it cannot follow the F (see ATTENTION below).
     Starting this time on the note F, let's take each fourth note:
F-7-C-7-G-7-D-7-A-7-E-7-B-(6)-(F)
          The F must be at the beginning because it cannot follow the B (see ATTENTION below).
Success!In both cases we have the same distance between the 7 notes:
     - at each third note, it is the distance of 5 semitones (the fourth);
     - at each fourth note, it is the distance of 7 semitones (the fifth);
     - notice that the 7 notes are placed in the same order,
          but in the inverse direction.

ATTENTION
In the first two cases, with variable distances of 1-2 and 3-4 (semitones),
we managed to make a loop and return to the original note, ready to start all over an octave or two higher.
On the other hand, in the two following cases,
if we wish to maintain our constant distance of 5 or 7 semitones, we cannot make a loop
because the distances F-B and B-F are both 6 semitones.

Extending the series of fifths
But we can do something else.
We can lengthen our series of notes, toward the right and toward the left,
     and discover the world of  sharp and flat notes.
We will use the second series where the notes are all a fifth (7 semitones) apart.
Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#

We have here 3 series of the notes F-C-G-D-A-E-B:
     once with the notes flat,
     once with the notes natural (a semitone higher), and
     once with the notes sharp (another semitone higher).

It is evident that there is a change in which the notes become
sharper toward the right and
flatter toward the left.

At first sight there seems to be a change of semitone as we pass from the flat notes to the natural notes and another as we pass from the natural notes to the sharp notes. But since the distance between Bb-F (and between B-F#) is exactly the same as elsewhere, there cannot be a sudden change at these points. We must therefore accept the inevitable conclusion: SHARPENING TOWARD THE RIGHT AND FLATTENING TOWARD THE LEFT MUST BE EQUAL AT EACH NOTE AND GRADUAL OVER THE ENTIRE SERIES. We call this process of gradual sharpening and flattening the dimension of Chrominicism.

     We could call:
C-D-E-F-G-A-B-C, the scale of Melody,
C-E-G-B-D-F-A-C, the scale of Harmony, and
F-C-G-D-A-E-B, the scale of Chrominicism.

You might enjoy a more graphic representation of Chrominicism called
Parking Lot