Now we know that each interval represents a definite ratio between the periodicities of its two tones. In the case of the Fifth the ratio is 2:3 (in the natural scale). This means that the lower tone receives its character from being related to the upper tone by the ratio 2:3. Similarly, the upper tone receives its character from the ratio 3:2. The specific character of an interval arising out of the merging of its two tones, therefore, is determined by the ratio of their ratios. In the case of the Fifth this is 4:9. It is this ratio, therefore, which underlies our experience of a Fifth.

The cosmic factor corresponding to the periodicity of the single tone in music is the orbital period of the single planet. To the musical interval formed by two tones corresponds the double ratio of the periods of any two planets. Regarded thus, Kepler's law can be expressed as follows: The spatial ordering of our planetary system is determined by the interval-relation in which the different planets stand to each other.

By thus unlocking the ideal content hidden in Kepler's third law, we are at the same time enabled to do justice to the way in which he himself announced his discovery. In textbooks and encyclopaedias it is usually said that the discovery of the third law was the surprising result of Kepler's fantastic attempt to prove by external observation what was once taught in the school of Pythagoras, namely, that (in Wordsworth's language):

'By one pervading spirit
Of tones and numbers all things are controlled.'

Actually, Kepler's great work, Harmonices Mundi, in the last part of which he announces his third law, is entirely devoted to proving the truth of the Pythagorean doctrine that the universe is ordered according to the laws of music. This doctrine sprang from the gift of spiritual hearing still possessed by Pythagoras, by which he could perceive the harmonies of the spheres. It was the aim of his school to keep this faculty alive as long as possible, and with its aid to establish a communicable world-conception. The Pythagorean teaching became the foundation of all later cosmological thinking, right up to the age which was destined to bring to birth the spectator-relationship of man's consciousness with the world. Thus it was left to Copernicus to give mankind the first truly non-Pythagorean picture of the universe.