History of the mathematician and statesman Archytas of Taranto

Only fragments of Archytas’ works have survived, but much has become known through other ancient authors who dealt with his works. So Archytas was convinced that a balanced and predictable distribution of property is a prerequisite for social peace in a society. He was generally loath to punish someone in anger, and preferred to refrain from punishment when judgment without emotion would be impossible. In his view, man reaches scientific knowledge by progressing from the general to the specific. Archytas also considered independent discovery to be more effective than learning from documented knowledge.

Early search for harmony and music

Archytas also devoted himself to music. The Pythagoreans examined the laws of musical sounds, which were seen as pleasant. The ratios of integers played an important role here: if you halve the string length in a monochord, a pitch higher than the major octave will be produced (ratio 1:6 = 2:1). If the length of the chord is shortened by a third, that corresponds to a fifth note above the root (9:6=12:8=3:2), shortening the length by a quarter results in a quarter (12:9)=8:6=4: 3).

The fact that the numbers 1, 2, 3, 4 of the Tetraktys appear here was for Pythagoras confirmation of the existing universal harmony. Also, if you divide the length of the chord into twelve equal segments, nine is the arithmetic mean of six and twelve, and eight is the harmonic mean of six and twelve (the term “harmonic” comes from Archytas).

In his theory of means and proportions, Archytas was able to prove that the geometric mean of two numbers cannot be rational if they are proportional n 🙁n + 1) Stand up. Therefore subdivision of pitch intervals is only possible by forming arithmetic or harmonic values.

Euclid continued Archytas’ music theory in his work “Sectio canonis”. One can also assume that Euclid’s genealogy (Books VII and VIII of the “Elements”) is based mainly on the texts of Archytas.

Archytas believed that sound is produced by the collision of moving objects and is transmitted through the air as pressure. Higher notes correspond to faster movement of the voice while lower notes correspond to slower movement.