Tetraktys
The Tetraktys ( Greek τετρακτύς tetraktýs " quartet " or "group of four") is a term from the numerology of the ancient Pythagoreans . He played a central role in Pythagorean cosmology and music theory, since the Tetraktys was seen as the key to understanding world harmony.
Ancient meaning
The Pythagoreans called Tetraktys the total of the numbers 1, 2, 3 and 4, the sum of which is 10. Since the ten ( Greek δεκάς dekás “number of ten”, “group of ten”) is the sum of the first four numbers, it was assumed that the four “generates” the ten. Ten already played a prominent role due to the fact that it served as the basic number of the decimal system for Greeks and “barbarians” (non-Greeks) alike . Moreover, as Aristotle reports, the Pythagoreans viewed the ten as “something perfect” that “embraces the whole essence of numbers” because of its connection with the Tetraktys. This is why ten was also called the “holy number”.
The Pythagorean cosmology was based on the assumption that the cosmos is harmoniously ordered according to mathematical rules. In this world interpretation, the tetraktys was a key term because it expressed the universal harmony. Therefore, some Pythagoreans assumed that there must be ten celestial bodies in motion, even though only nine were visible - a speculation that resented Aristotle.
The discovery of world harmony was attributed to Pythagoras of Samos , the founder of the Pythagorean tradition. Therefore the Pythagoreans had an oath formula that read:
"No, with him who gave our soul the Tetraktys, which contains the source and root of the ever-flowing nature."
The person who gave the Tetraktys was Pythagoras.
In the " Golden Verses " (carmen aureum) , a poem popular in antiquity and then again in the Renaissance, which summarized the Pythagorean teachings, there is a slightly different version of the formula (verses 47 and 48):
"Yes, with him who gave the Tetraktys to our soul, source of ever-flowing nature."
The tetraktys was expressed with counting stones (psēphoi) by placing the four numbers one above the other in the form of an equilateral triangle . There was also a symbolism here, since the equilateral triangle was considered a perfect figure.
In music, the Pythagoreans established that the basic harmonic consonances fourths , fifths and octaves , which are represented by the numerical ratios 4: 3 (= 8: 6), 3: 2 (= 9: 6) and 2: 1 (= 12: 6) with the four numbers of the tetraktys can be expressed, as well as two further intervals: the duodecime consisting of octave and fifth (3: 1) and the double octave (4: 1). Only these five intervals were recognized as a symphon. The undezime (8: 3), which does not fit into the framework of the Tetraktys, was therefore excluded from the consonant intervals due to a theoretical consideration, although it is perceived as consonant or at least not as dissonant. The theory of the Tetraktys took precedence over sensory perception. This approach was criticized by the empirical music theorist Ptolemy .
In addition to the group of numbers one to four, the Pythagoreans had other significant groups of four numbers, which were also called tetraktys. In music theory - as is also passed down in the legend of Pythagoras in the forge - the group 6, 8, 9, 12 was particularly important, as these numbers were assigned to the unchangeable strings of the lyre (Hypate, Mese, Paramese, Nete). The music theorist Nicomachus of Gerasa calls this group the “first” Tetraktys, whereby “first” is to be understood in terms of rank. It states that the six corresponds to the lowest note, the hypate, and the twelve to the highest, the nete.
In geometry, too, the four elements point, line (length), surface (width) and physicality (depth) were found to be a quaternary, which for the Pythagoreans pointed to the Tetraktys. The point was assigned to one, length to two, area to three and physicality to four.
The Jewish scholar Philon 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 Pythagorean consonance theory based on the Tetraktys concept largely shaped medieval music theory. The divergent view of Ptolemy was also known, as the late ancient scholar Boethius had presented it in the fifth book of his work De institutione musica . The question of including the undezime in the group of consonances was discussed controversially, with the Pythagorean view that this interval is not consonant prevailing.
Modern reception
In his work De coniecturis (1440), Nikolaus von Kues took the view that there is harmony in the numbers 1, 2, 3 and 4 and their combinations; but he did not refer explicitly to the Pythagorean tradition. The humanist Johannes Reuchlin compared in his work De verbo mirifico (About the miraculous word) , published in 1494, the tetragram, which represents the divine name YHWH , with the tetraktys. Raphael reproduced it on a panel in his fresco The School of Athens . Even Johannes Kepler has in his 1619 published work Harmonice mundi concerned ( "Harmony of the World") with the Tetraktys.
Relation to Pythagorean triples
Starting from the (degenerate) primitive triple (1,0,1) a tetraktys (of four operators) forms the root (3,4,5) known for the calculation of all further primitive Pythagorean triples (x, y, z).
literature
- Charles H. Kahn: Pythagoras and the Pythagoreans . Indianapolis 2001, ISBN 0-87220-576-2 , pp. 31-36, 84f.
- Bartel Leendert van der Waerden : The Pythagoreans . Zurich 1979, ISBN 3-7608-3650-X , pp. 103-109.
- Paul Kucharski: Etude sur la doctrine pythagoricienne de la tétrade . Paris 1952.
- Armand Delatte: Etudes sur la littérature pythagoricienne . Paris 1915, pp. 249-268 (chapter La tétractys pythagoricienne ).
- Theo Reiser: The Secret of the Pythagorean Tetraktys . Lambert Schneider, Heidelberg 1967.
Web links
- Tetraktys interpretation by Holger Ullmann
Remarks
- ^ Walter Burkert: Wisdom and Science . Nuremberg 1962, p. 64.
- ↑ Aristotle: Metaphysics 986a8-10.
- ↑ Bartel Leendert van der Waerden: The Pythagoreans . Zurich 1979, pp. 457f.
- ↑ Aristotle: Metaphysics 986a10-15.
- ^ Leonid Zhmud : Science, philosophy and religion in early Pythagoreanism . Berlin 1997, pp. 184f. One of the main sources is Sextus Empiricus , Adversus Mathematicos 4, 2–9.
- ↑ Barbara Münxelhaus: Pythagoras Musicus . Bonn 1976, pp. 22-24, 26-28, 41, 71, 84f., 110, 185-191.
- ^ Sextus Empiricus: Adversus Mathematicos 4, 4–6.
- ↑ Barbara Münxelhaus: Pythagoras Musicus . Bonn 1976, pp. 88-94.
- ↑ De coniecturis II.2 (83); see Werner Schulze: Harmonics and Theology with Nikolaus Cusanus . Vienna 1983, p. 70f.