The Pythagoreans' beautiful fallacy

-December 26, 2013

Note: Thumbnail image of Pythagoras is from the University of California, Riverside Physics & Astronomy website.

PythagorasThe Greek philosopher, Pythagoras, founded a school around the fifth century B.C. Pythagoras and his students/followers believed that the universe could be understood in terms of whole numbers, but they never succeeded in proving their premise. They developed this belief by observing certain areas such as music, astronomy and mathematics.

Aristotle commented that, “They supposed the elements of number to be the elements of all things, and the whole of heaven to be a musical scale and a number”

The octave was represented as a ratio of 2:1 because if the length of a musical string is halved, it is musically one octave higher. This was one of the first discoveries that did support their belief. Furthermore, the ratio of 3:2 corresponds to a fifth and 4:3 is a fourth. A single tone was the difference between 5th and a 4th, and was therefore 9:8 which is 3:2 divided by 4:3.

But the task of constructing an entire scale is extremely complex. This even challenges today’s musicians. All possible solutions are merely approximations. It is not possible for a fixed scale, like a piano possesses, to include all the perfect 5ths and 4ths as a singer or songwriter would ideally want. The solution that divides an octave into 12 equal tones never has any of them perfectly correct.

The Pythagoreans experimented1 with plucked strings and found that the intervals that pleased one’s ears were: the octave (1:2), the fifth (2:3) and the fourth (3:4). If we add two Greek composite consonances of the octave plus fifth (1:2:3) and double octave (1:2:4), they discovered that all the musicals that they thought were beautiful, these five sets of ratios were all composed of the simple integers 1,2,3 and 4. These were the numbers in the sacred tetractys which added up to the number of fingers a human possesses. They thought that they had discovered a basic law of the universe.

Pythagoras also believed that the distances between the planets had the same ratios as the harmonics of the plucked string. That meant that the solar system consisted of ten spheres revolving around a central fire (sun) while each sphere gave off a sound as a projectile would as it flew through the air. The closer spheres had lower tones and the further moved faster at higher pitched tones. This gave the harmony of the “Music of the Spheres”

“…and the whole heaven to be a musical scale and a number…”

Beautiful thought, but later to be found incorrect.


1 Darthmouth website mathematics section

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