The big surprise of Chrominicism
The 7 notes which we call natural are not equally natural -
the D, in the center, is the only perfectly natural note;
the G is slightly flat, the C, more so and the F, even more so;
the A is slightly sharp, the E, more so and the B, even more so.
To clarify this notion of "more or less sharp" (or flat),
let's imagine a parking lot where we line up cars,
with white lines delimiting the space allowed to each car.
Now let's remove the cars and replace them with bicycles
which, for all intents and purposes, have no appreciable width.
Each bicycle is now free to park
(a) in the center of its allotted space,
(b) more or less to the left, or
(c) more or less to the right,
but always inside the allotted space.
If we transpose this parking concept to the Chrominic Position of the notes
(their degree of flatness or sharpness),
we can say that a note is placed more or less
above (sharp) or
below (flat) the center of the space allotted to it.
In the drawing on the left we see a column of 7 separate diagrams,
one for each note, which shows:
the note D, in the center, precisely at the center of its allotted space;
the note A, above, is placed 1 “notch” above the center (sharp),
the note E is placed 2 notches above the center, and
the note B is placed 3 notches above the center,
the note G, below, is placed 1 notch below the center (flat);
the note C is placed 2 notches below the center, and
the note F is placed 3 notches below the center.
The Chrominic Position of a note is the same at all octaves.
All the Cs are placed 2 notches lower,
all the Ds are neutral,
all the Es are placed 2 notches higher ...
What, precisely, is the size of each notch?
It is very difficult, if not impossible, to determine this size
by comparing two notes of different names,
but between two notes of the same name, it is much simpler.
If we start from the note F and go up 7 fifths, we reach the note F#
which we know to be a semitone sharper than the F.
Thus, if after 7fifths we sharpened 1 semitone,
we sharpened 1/7 of a semi-tone (1/84 of an ocatve) at each fifth.
Why are the semitones placed
between E-F and B-C
and not elsewhere?
Let's build a complete parking lot and place each note
according to its Chrominic Position, as found in the previous drawing.
In the drawing on the left we see a complete parking lot,
covering an octave from C to C,
divided into 84 notches,
and, at every 12 notches, the center of the allotted space for each note.
Starting from the bottom:
the note C, placed 2 notches flat, according to its Chrominicism;
the note D, placed dead center, the only truly natural note;
the note E, placed 2 notches sharp;
the note F, placed 3 notches flat;
the note G, placed 1 notch flat;
the note A, placed 1 notch sharp,
the note B, placed 3 notches sharp, and
the note C again, placed 2 notches flat.
the same Chrominic Position for both Cs.
Here we see why, with this phenomenon of Chrominicism,
we have whole tones and semitones in the scale.
The 5 whole tones (14 notches in size) are placed
where the lower note is flatter (or less sharp) than the upper note
(increasing the size of the interval by 2 notches)
between C-D, D-E, F-G, G-A, and A-B.
The 2 semitones (7 notches in size) are placed
where the lower note is sharp and the upper note is flat,
(decreasing the size of the interval by 5 notches)
between E-F and B-C.
And why do we have precisely 5 whole tones and only 2 semitones?
Because a whole tone appears between notes that have
a difference of Chrominic Position of 2 notches,
and this occurs 5 times,
and a semitone appears between notes that have
a difference of 5 chrominic notches,
and this occurs only 2 times.