If we place
     the 7 flat notes,
     the 7 natural notes, and
     the 7 sharp notes in the order of perfect fifths,
it seems we have clearly the impression of a three-layered stair-case, with
     the flat notes on one level,
     the natural notes on another level a semitone higher, and
     the sharp notes on another level a semitone higher still, thus:

But this is not possible
     because the fifth from Bb to F is constructed on the same 3:2 ratio as all the other fifths
          and cannot produce an exceptional change of level.
     This also applies to the fifth from B to F#.

The only solution
The only possible alternative to explain the sharpening as we proceed to the right
     is that the sharpening is perfectly gradual.

The consequences are inevitable

The 7 natural notes are not equally natural
- the D, in the centre, is the only perfectly "natural" note;
- the G is slightly flatter (1 "notch");
- the C is slightly flatter still (2 "notches"), and
- the F is the flattest of the three (3 "notches");
the same applies to
- the A (1 "notch"),
- the E (2 "notches"), and
- the B (3 "notches") which are progressively sharper.

What is the size of these notches?
How much sharper (or flatter) are notes with respect to each other?
It is very difficult to establish the difference between two adjacent notes in the series.
However, between the notes F and F# (7 fifths apart), the difference is obvious - 1 semitone.
If it takes 7 fifths to sharpen 1 semitone, each fifth sharpens 1/7 of a semitone
     (1/84 of an octave), the size of each "notch".

These calculations were made with equi-tempered tuning. They would be more complicated with Just Intonation but the proof of the existence of CHROMINICISM would be equally valid.

Another proof

It is actually possible to use the "staircase" concept as supplementary proof.
1. Especially with equi-tempered tuning, all keys have exactly the same relationships between the degrees of the scale.
2. In the key of G major, we have only one sharp note, the F#. It is degree VII and it is the sharpest of the seven.
     It follows that degree VII must also be the sharpest degree in the Key of C, namely the note B.
3. In the key of F major, we have only one flat note, the Bb. It is degree IV and it is flatter than the others.
     Degree  IV must also be flatter than the others in the Key of C, the note F.
4. And so on ...

Tones and Semitones

Now let's look at the scale
Why do we have these whole tones and semitones?
     Each whole tone (C-D, D-E, F-G, G-A, A-B)
          has a lower note which is flatter (or less sharp) than the upper note,
               producing a larger interval.
     Each semitone (E-F, B-C)
          has a lower note which is sharper than the upper note (which is flatter),
               producing a smaller interval.
          There are only 2 places in the scale where this happens
               and this is where the semitones are located.

Hearing Chrominicism

Some will complain that they cannot "hear" or "feel" this gradual change of sharpness and flatness, but we cannot really "feel" the rotation of the earth either. That does not prevent these two concepts from explaining a lot of phenomena. On second thought, is it really that difficult to "feel" the chrominicism?

Let’s make a few practical tests
1. Sing the notes C-D-E.
     We have here one of the most fundamental melodic groupings,
          degrees 1, 2, 3 of the Major Scale.
     Twice we have gone up a whole tone (major second)
          2 notches of sharpness each time, 4 notches in all between C and E.
               So far so good.
2. Now let's add another whole tone and sing C-D-E-F#.
     This proves appreciably more difficult,
          the natural tendency to sing F, instead of F#, being quite strong. Why?
     Because we are always using the note C as a point of reference and
          the 6 chrominic notches between C and F#
               are far greater than the 1 notch between C and F.
3. As we try to sing C-D-E-F#-G#,
4. and even worse C-D-E-F#-G#-A#,
     things become more and more difficult, and
          the normal tendency being to always sing a natural note,
               a semitone away from the preceding note,
          rather than a sharp note, a whole tone away.

In other words the "Chrominic Spread" between the first and last notes
(8 notches between C and G# and 10 between C and A#)
is very noticeable.

Let's try something else, starting on the note B -
1. Sing the notes B-C-D-E
     No problem.
2. Sing the notes B-C-D-E-F#.
          Still no problem, because the point of reference is now the note B (instead of C)
               and the "Chrominic Spread" between B and F# is only 1 notch.
Once on the F#, the rest is relatively easy.
3. Sing the notes B-C-D-E-F#-G#-A#-B.

Once the note B is established as the point of reference -
the notes C, D, E, are respectively 5, 3, 1 notches flatter, and
the notes F#, G#, A#, are respectively 1, 3, 5 notches sharper,
an acceptable Chrominic Spread, even for the relatively untrained ear.

The following "layman's" presentation will not add very much to what precedes,
but you might find it fun to see CHROMINICISM presented
in a slightly different way.
It's entitled
Who's Afraid Of The Big Bad Scale?
You might also enjoy a more graphic representation of Chrominicism called
Parking Lot