First exercise book in Harmony – lesson one

Weissenfels 11 maggio 1900

Today we started harmony and Kretzschmar has informed me that he intends to develop his dissertation from the early steps, even if - at the beginning, it may happen that I hear things I deem to know well.
"In order to test its solidity, I want to show you the logic of the concatenation by which the system develops in its early steps," he told me, and I readily agreed on his plan which did not want to take anything for granted, be already known or test my fragmentary musical culture.

"To begin demonstrating the System we must define the material with which we intend to work.
The octave interval, acknowledged by physics, nature and the ear, can be our first unit of reference.
It defines an interval after which the voices contained within will repeat on a lower or higher register.
In order to develop a system, as we are taught by positive thought, it is convenient to divide such intervals in equal parts, so as to define the common multiple minimum, the smallest unit that we will have at our disposal to build a complex structure.
The System that I want to illustrate divides this octave interval into 12 equal parts, defining 12 different voices, each the same distance from each other.


"You will excuse the superficiality with which I expose this part of the dissertation, neglecting the development of the tempered scale, but I want to reach music and its exercise as soon as possible."

This is how the Maestro specified, why he was excusing himself for using a shortcut that I have since found convenient and useful.
In the end, the tempered scale is a correction that applies to the natural one, therefore I understand the rush, on Kretzschmar's part, to close a topic that weakens the concept of natural derivation of the system and loosens it from its foundations. Now a sentence from one of his lectures gains meaning, one in which he admitted that Nature is full of signs hinting at an uncertain world, one which today he did not want to repeat, thereby weakening the idea of the solidity of the system he wants to impart to me.

These 12 voices a semitone away from each other are the smallest unit by which we can work. The semitone is the shortest interval that we can use to make a voice rise or sink.

The 12 notes of the chromatic scale

Rising on the keyboard from the bass to the high tones and moving from left to right, the black keys will be called # (sharp). Descending from high tones to low ones, the black keys will be called b (flat).
From the physical composition of each single sound the Major scale emerges forcefully after the chromatic one, composed of all the harmonics of each sound.

It was Claudius Ptolemy an Egyptian born in Alexandria who named the best of all known scales, the Major scale, once called the Ionic scale.

The Major scale of C is well represented by the piano keyboard that one can see above: the notes composing it are all from the white keys. Even on the piano keyboard it is easy to see that between E and F, as with between B and C there are no black keys, that is, there are no intermediate notes separating them.
That is why in the C Major scale, the interval between E and F and that between B and C are called diatonic semitones, when pertaining to the scale of C Major (while the intervals C-C#, D-D#, A-Ab, etc. are chromatic semitones).

As evidenced on the staff above, the architecture of the Major scale is determined by a sequence of intervals of tone and semitone. That sequence is TTSTTTS (Tone and Semitone) as easily verified on the piano keyboard above.
This architecture allows us to build a Major scale on each of the 12 notes of the chromatic scale. That is to say that taking each of the 12 notes of the chromatic scale as the first note, we can build 12 different Major scales.
No more rough material. This is already semi-finished and very useful for the building of our System.
If we want to write a Major scale in E, the first tone will be E and the others will be at a distance of TTSTTTS, keeping the chromatic scale as a reference. That is:

At this point, Kretzschmar found it indispensible to demonstrate how every single note containsall other notes within its harmonics in the Major scale. He then also wanted to define what harmonics are and to what extent they define a note.
I will not report on that part of the "dissertation". It is, indispensible, perhaps, to demonstrate that this system derives directly from Nature and builds itself on its own principles, but useless to the practical needs that spawned this notebook.
Unfortunately, the remainder of the lesson veered towards acoustic philosophy.

The Master ended the lesson by gravely emphasizing how the Major scale is the spring from which our modal system flows. Therefore the assignment received is to establish the Major scales of the 12 tones in the chromatic scale and obviously learn to play them on the piano with ease.