The following are among the Pythagorean doctrines taught at his school:
Pythagoras believed that the universe was formed by the harmonies of the Limited and the Indefinite, Form and Matter, and other such opposites. The full Table of Opposites appears below:
The ordering of the table of opposites is not random, but is based on the relationships that Pythagoras found between things on each side. For instance: if successive gnomons (carpenter's squares) are placed around a single object (One), there is room for an odd number of similar objects in the spaces between each gnomon (Odd), and the final result is a Square. However, if the first gnomon is placed around two objects (Plurality), an Even number of objects fits in each space, and the final result depends on the placement of all the previous objects and is irregular (Oblong).
The typical concept of "number" is that of a thing to be used, a means to an end. Pythagoras viewed it instead as a living thing, something which was not invented but discovered through experiment, a physical quantity like the motions of the heavens. It was divine and universal, and its reach went beyond the physical dimensions into the sacred dimension of the mind.
One, the Monad, was the principle of Unity from which all things begin. Two, the Dyad, was Duality. It was both the beginning of strife and the possibility for the development of relationships between things, as everything is no longer the same. The Triad, Three, forms a bridge, allowing Relation between the two extremes. A graphic example of this is shown below.
The Pythagoreans had four branches of Number: Arithmetic consisted of solely Number, Geometry was Number combined with space, Music was Number in time, and Astronomy was the mixture of Number, space, and time.
Pythagoreans liked to express numbers as geometrical figures; the foremost of these is the Tetraktys (shown below). It holds meaning both numerological and musical, and unifies several of the Pythagorean Opposites.
Starting at Unity (one), the Tetraktys proceeds through the ordered numbers two, three, and four, which add up to a second Unity, ten. These four numbers also correspond to the ideas of a point, line, plane, and three-dimensional surface. As we will see in the following section, they also form the basis for musical scales: the ratio of 1:2 is an octave, 2:3 the perfect fifth, and 3:4 the perfect fourth. This figure encompasses both Odd and Even, Square and Oblong, and so the Tetraktys (or Decad) was often called the Kosmos (world order), Ouranos (heaven), or Pan (the All.)
Pythagoras began his experimentation in music with a monochord, an instrument consisting of a single taut string with a movable bridge for tuning. By halving the length of the string, the frequency of the sound is doubled -- in other words, the note produced is an octave higher than the first one. The ratio of the octave is thus 1:2.
Consider the octave 6:12. Taking the arithmetic mean of these numbers (adding them together and then dividing by 2) yields the number 9. The ratio 6:9, or 2:3, is a perfect fifth. Similarly, the "harmonic method" (multiplying the extremes, doubling the result, and dividing that by the sum of the two extremes, expressed as 2AB / A+B) yields the number 8. This proportion, 6:8, is the perfect fourth. These two operations are the foundation of the harmonic scale. The rest of the notes making up the octave can be found as follows:
The base of the octave is C. This can be increased to D by a change in string length of 8:9. D goes to E in another increment of 8:9. The ratio of E to F is 243:256; this amount was called the "leimma" or "left over." One can move from G to B with more increments of 8:9; the final step from B to C is the leimma again.
Pythagoras used the mathematical order to music as an example of how Number pervades the universe. Surely, if it could be used to express music, it could be related to all other things. He was also said to have used music for healing or soothing purposes, playing certain harmonies and sequences of sounds to calm people's anger or arouse them from sleep in the morning.
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