The quotes from a textbook Deep harmonization. 


We do not know the beginnings of music. Myths of many nations attribute a divine origin to music. In early days music really belonged to a cult area, only late on it becomes the art of expression. While looking for the beginnings of music we also have to take into account various phenomena from sound field. West European notion of music derives from Greece antiquity and from big ancient cultures of the Near and the Far east.

PYTHAGORAS from Samos (570-496 A.C.) is considered to be a "founder" of music to the effect that he gave an explanation of mathematical foundations of music (=tonal system).

He supported his explanation by acoustical measurements on a monochord.

Foundation stones and numeral proportions of his system:

perfect octave, 1:2

perfect fifth, 2:3 (=slightly higher than a tempered fifth)

perfect fourth, 3:4 (= a complementary interval, slightly lower than tempered fourth, it produces octave together with fifth).

These proportions can be expressed via numbers 6, 8, 9, 12, when we have two fifths 2:3, two fourths 3:4 and one whole tone 8:9 within the octave; the octaves, fifths and fourths are perfect in the Pythagorean tuning, they are perfect consonances.

Other intervals derive from the ones mentioned earlier, for example the third from two whole tones 8:9, a semitone by subtracting of two whole tones 8:9 from the fourth 3:4, their numeral proportions are difficult and therefore discordant.

Within Pythagorean tuning these tonal systems (rows) are possible:

  • The pentatonics without semitones is created by piling up of five fifths:

c-g-d-a-e, arranged within one octave: d-e-g-a-c [c-d-e-g-a]

  • The diatonics of seven tones: f-c-g-d-a-e-h, that is d-e-f-g-a-h-c [c-d-e-f-g-a-h]
  • The chromatics of twelve tones: but here arises a well known problem when the twelfth fifth is by so called Pythagorean comma (=23,5 cents) higher than the seventh octave with which it should be identical, and this difference repeats in all (from tempered tuning point of view) so called enharmonic tones, for example fis-ges, b-as and so on.

Pythagoras also formulated an idea about spherical harmonics: music as a component and at the same time a secret code of cosmic order - also movement of the cosmos and of the human soul is based on the same numeral proportions as interval proportions in music. He also formulated an idea about a musical ethos, that is an idea about he impact of music on a human being, on physical as well as mental sides of his existence - music is in its numeral nature a reflection of world layout, it also influences retroactively the mind and character of a human being, it becomes moral and social factor which pays role in education and social life.

His concept of spherical music reveals that each planet, star, plant, see or rock moves in specific rhythm and resonates with specific vibration. Many vibrations are founts of ancient wisdom which is sealed up in mantras and mandalas. Each thing in the universe, even the universe itself, constantly resonates in a certain rhythm.

Pythagoras called this entire rhythm spherical music and expressed its components via numbers.