Third lesson: the circle of fifths.

"In the preceding lessons we defined the chromatic scale,
the Major scale and all the intervals (harmonic or melodic).

We know how to build a Major scale on all the 12 notes of the chromatic scale,
thereby obtaining 12 Major scales.

This procedure defines the tonal system, 12 Major scales built upon the 12 notes of the chromatic scale, which we will soon see distinguished from the modal system, or, more precisely, bimodal,
since it builds on two scales:
a Major one (the ancient Ionic mode), which we have already seen,
and a Minor one (the ancient Eolic mode), which we shall soon discuss.

From every Major tonality we can obtain the relative Minor tonality and vice versa.

The relative Minor tonality builds upon the sixth degree of the Major one. The relative Major builds upon the third degree (unaltered) of the Minor one.

This implies the definition of 24 different tonalities,

12 Major and
12 Minor.

Once those 24 scales are established,
the development of the musical system proceeds in defining the relation between them.

The first feature that we can assign to this relation is that of similitude, that is arranging them on the basis of how many notes share a tonality with the other.
So, to recap, I advise you to place the 12 major tones on a chart and order them by similitude.

Here is how to order them:
Having defined the Major scale of C,
if we use the perfect 5th. (G) as the tonic to build another Major scale,
the tonality reached will only have one note (out of seven) distinguishing it from the tonality of C,
while the other 6 notes defining it are present in both tonalities.

And leaving me with just this rule as help,
he waited for me to arrange all the 12 Major scales
(obtained from the notes of the chromatic scale),
in order of similitude and proximity.

Here is how to order them:

Having defined the Major scale of C, if we use the perfect 5th. (G) as the tonic to build another Major scale, the tonality reached will only have one note (out of seven) distinguishing it from the tonality of C, while the other 6 notes defining it are present in both tonalities.

And leaving me with just this rule as help, he waited for me to arrange all the 12 Major scales (obtained from the notes of the chromatic scale), in order of similitude and proximity.

Below I added the table which I have created by arranging all the 12 notes' of a perfect 5th intervals, according to the rule expressed by Kretszchmar.
The only intervention by the Maestro,
who was impatiently hopping around as I compiled the table which I have placed above,
was that once arrived to the scale of B to build its subsequent Major scale,
he advised me to use the corresponding enarmonic of its perfect 5th.

Therefore, instead of F# Major,
which would have resulted in many double #,
its equivalent enarmonic Gb was used.

And then he again suggested highlight the different notes from the scale of C Major.

I have to say that when I gave him the table he observed it for a long time.
After, he wanted to hang it in front of the small upright piano
in the music study hall where he gives lessons.

He said that it will be very useful in building and interpreting the chords.
He then wanted me to observe some interesting things, which I show below, based upon my recollections, since his observations were many and very articulated and his rhetoric enthusiasm overwhelmed with passion.

"This table is the heart of the System.
From here we start building everything else.
This list defines all Major scales, then orders them by similitude.

The nearest ones will have just one note differing between them, then 2 notes, 3 notes and so on.
This table also indicates which notes are shared between one scale and another.

When composition time will come, it will be very useful to know that between the scale of Cmajor and that of B Major, for example, there are only 2 notes in common (B and E).

The other obvious observation is that neighboring scales can be ordered starting with the perfect 5th., following the clockwise direction in the circle of fifths (see below), or by perfect 4th., following the counterclockwise direction of the table.

Both these intervals lead to a Major scale which has just one sound distinguishing it from the one we started from.

For instance the scale of C and that of G (the 5th.) have only one differing note (Bb).

These last considerations are graphically represented by the circle of fifths
that I am about to introduce to you, and for which I request your undivided attention."

Then, out of a drawer the Maestro pulled a sign he had prepared
fore one of his lectures at the Society of Common Good;
he showed it to me with great enthusiasm and I reproduce it below.

"Graphically, the circle of fifths can be represented
as the face of a clock with the 12 tonalities in place of the hours.

Clockwise, they precede at the distance of 5 ascendants.
Counterclockwise they precede at a distance of 5 descendents (or 4 descendents).

This graph also shows the key signature defining the scale and its corresponding tonality.
Next to the Major tonality the relative minor mode of that tonality is also specified,
which builds upon the sixth degree of the Major one.

The graphic representation of the circle of fifths
inevitably leads one to think of the 12 tonalities as the hours on a clock
and the 7 notes as the days of the week.

Then again if the constitutive material of painting is space,
that of music is time and over this hypothesis I would like to quote a few arguments…"

This last thought took a more speculative direction, wherein the Maestro, as is in his nature, went all the way: "Music is our way to turn time into sound."
And I listened with great pleasure to the eccentric deductions and hypothesis of the Maestro.
Through his passion he managed to make you listen to music when he talks about it,
he manages to show it to you and also keep you in good spirits.