1 Historical precursors: Harmony and the cosmos in Pythagoras and Kepler

I would also like to consult the ear on this, though in such a way that the intellect articulates what the ear would naturally have to say. (Johannes Kepler)[1]

The first example of sound being linked with mathematically formulated knowledge and a resulting explanation of the world can be found in antiquity in Pythagoras of Samos and the one-stringed monochord. By shifting the position of the instrument’s bridge, one can demonstrate in a simple way the connection between string length and pitch or frequency relationship, because when the length of the string is reduced by half, the tone produced rises by one octave and its frequency doubles. Based on his observations, Pythagoras determined that proportions and intervals or numbers and tones are inseparably connected.[2] Hence, simple numerical proportions result in musical harmony, which for Pythagoras was proof that the laws of nature are based on a specific harmony that can be experienced audibly. Pythagoras thus furnished one of the few early examples of Western thought in which a sonic frame of reference has explanatory authority. This is remarkable, because philosophical insight — the word itself suggests it — is more frequently related to the sense of sight; one need only call to mind Plato’s cave allegory.

Pythagoras’s numerical proportions can also be found in various other early scientific investigations. In his book Harmonices Mundi, the mathematician and astronomer Johannes Kepler assigns tone series or intervals to the planets, which he calculates based on the relationships between distance from the sun, orbit, speed, and time, and in this way arrives at a music of the spheres based on numbers. His discovery that the speed of the planets, if one measures them at the two points of their elliptic orbit [that are farthest away from one another] … form interval proportions,[3] can be traced in musical notation.

While Pythagoras starts from an experience of sound based on an experiment and uses this to explain the world, Kepler imagines sound based on a model of the movement of the planets. Kepler’s acoustic laws also probably served to prove that the harmonies of the cosmos are based on a higher (divine) intention.