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| The Pythagorean Oath mentions the Tetractys: "By that pure, holy, four lettered name on high, nature's eternal fountain and supply, the parent of all souls that living be, by him, with faith find oath, I swear to thee." Credits: Michæl Paukner |
Albert Mackey (1873) - "The Greek word 'Tetractys' signifies, literally, the number four, and is synonymous with the quaternion; but it has been peculiarly applied to a symbol of the Pythagoreans, which is composed of ten dots arranged in a triangular form of four rows."
The first four numbers symbolize the harmony of the spheres and the Cosmos as: (1) Unity (Monad); (2) Dyad - Power - Limit/Unlimited; (3) Harmony (Triad); (4) Kosmos (Tetrad) - The four rows add up to ten, which was unity of a higher order (The Dekad). The Tetractys symbolizes the four elements — fire, air, water, and earth. The Tetractys represented the organization of space: the first row represented zero dimensions (a point). The second row represented one dimension (a line of two points). The third row represented two dimensions (a plane defined by a triangle of three points). The fourth row represented three dimensions (a tetrahedron defined by four points).
The Pythagorean musical system is based on the Tetractys as the rows can be read as the ratios of 4:3 (perfect fourth), 3:2 (perfect fifth), 2:1 (octave), forming the basic intervals of the Pythagorean scales. That is, Pythagorean scales are generated from combining pure fourths (in a 4:3 relation), pure fifths (in a 3:2 relation), and the simple ratios of the unison 1:1 and the octave 2:1. The diapason, 2:1 (octave), and the diapason plus diapente, 3:1 (compound fifth or perfect twelfth), are consonant intervals according to the tetractys of the decad. The diapason plus diatessaron, 8:3 (compound fourth or perfect eleventh), is not.