Tetractys

The Tetraktys (Greek τετρακτύς tetraktýs " tetrad " or " group of four " ) is a term used in the theory of numbers the ancient Pythagoreans. He played in the Pythagorean cosmology and music theory a central role, since you saw the key to understanding the world harmony in the Tetraktys.

Ancient importance

Tetraktys as the Pythagorean indicated all of the numbers 1, 2, 3 and 4, the sum of which results in 10. Since the Ten ( Greek δεκάς DEKAS " number ten ", " Group of Ten ") is the sum of the first four numbers, it was assumed that the Tetrad "created" the Ten. The Ten came already by the fact that they served equally among the Greeks and "barbarians" ( non-Greeks ) as the basic number of the decimal system, a prominent role. From the Pythagoreans the Ten was also, as Aristotle says, considered due to their connection with the Tetraktys as " something perfect ," which " includes the whole essence of numbers." Therefore, the Ten was also called " sacred number ".

The Pythagorean cosmology was based on the assumption that the universe according to mathematical rules is arranged harmoniously. In this world, the interpretation Tetraktys was a key concept because it expressed the universal harmony. Therefore, some Pythagoreans assumed that there must be ten moving celestial bodies, although only nine were visible - a speculation which resented them Aristotle.

The discovery of the cosmic harmony was Pythagoras of Samos, the founder of the Pythagorean tradition attributed. Therefore, it was with the Pythagoreans an oath, which was:

"No, by Him who has the soul pass the Tetraktys containing the source and root of the eternally flowing nature. "

With the person who handed the Tetraktys, Pythagoras was meant.

In the " Golden Verses " ( carmen aureum ), one in antiquity and then again popular in the Renaissance poem that summarized the Pythagorean doctrines, is a slightly different version of the formula (verses 47 and 48):

"Yes, in the person who has passed the Tetraktys our soul, source of eternal flowing nature. "

The Tetraktys was expressed with Zählsteinen ( psēphoi ) by the four figures were arranged in the form of an equilateral triangle above the other:

This, too, was a symbolism since the equilateral triangle was regarded as a perfect figure.

In music, the Pythagoreans discovered that the harmonic Grundkonsonanzen fourth, fifth and octave, where the ratios 4:3 ( = 8:6 ), 3:2 ( = 9:6 ) and 2:1 ( = 12:6 ) have been assigned the Tetraktys can be expressed with the four figures, as well as two other intervals means the combination of fifth and twelfth octave (3:1) and the double octave (4:1). Only these five intervals were recognized as symphon. The eleventh ( 8:3 ) that does not fit into the framework of Tetraktys was therefore excluded from the consonant intervals based on a theoretical consideration, although it is at least not perceived as consonant or dissonant as. The theory of Tetraktys had priority over the sensory perception. This approach was criticized by the empirically -minded music theorist Ptolemy.

In addition to the group of numbers one through four were among the Pythagoreans are other important groups of four numbers, which were also called Tetraktys. In music theory was - is delivered as well as in the legend of Pythagoras in the forge - the group 6, 8, 9, 12 is particularly important because these numbers were assigned to the immutable strings of the lyre ( Hypate, Mese, Paramese, Nete ). The music theorist Nicomachus of Gerasa this group are therefore referred to as the "first " Tetraktys, where " first" is outranked to understand. It indicates that the six equivalent, the lowest note, the Hypate the Twelve the highest, the Nete.

Also in the geometry found himself with the four elements point, line (length ), area (width) and physicality (depth) a tetrad, which indicated for the Pythagoreans to the Tetraktys. The point of the One, the length of the two, the area of ​​the Three and the physicality of four was assigned.

The Jewish scholar Philo of Alexandria used the Tetraktys concept in commenting on the book of Genesis. He related it to the creation of the stars on the fourth day of creation.

Middle Ages

The foot on the Tetraktys concept Pythagorean Konsonanzlehre shaped the medieval music theory largely. The dissenting opinion of Ptolemy has also been known since the late antique scholars Boëthius had explained it in the fifth book of his treatise De musica institutione. The question of the inclusion of the eleventh in the group of consonances was discussed controversially, the Pythagorean conception prevailed.

Modern reception

Nicholas of Cusa advocated in his treatise De coniecturis ( 1440) considered that in the numbers 1, 2, 3 and 4 and their combinations there all harmony; but he did not appeal explicitly to the Pythagorean tradition. The humanist Johannes Reuchlin compared in his 1494 published work, De verbo mirifico ( via the wonder-working word) the Tetragrammaton representing the name of God, YHWH, with the Tetraktys. Raphael gave it to his fresco The School of Athens on a blackboard again. Johannes Kepler also has in his 1619 published work Harmonice mundi ( " The Harmonies " ) deals with the Tetraktys.