In this book, it is impossible to thoroughly discuss Kepler’s Harmonice Mundi—the work that was most important to him, and a treasure in many ways for modern harmonics. After the completion of this textbook, I published a more thorough digest of the content of Harmonice Mundi in the Schweizer Rundschau (Oct.-Nov. 1946, pp. 545-553).
What could be said in summary I attempted to say in Hörende Mensch, p. 171 ff. The plea there for a new collected edition of the works of this genius has been answered in the excellent translations of Prodromus, Marswerk, and Harmonice Mundi, by Max Caspar (see the Bibliography for this chapter). These new editions, revisions, and translations give us the duty as harmonists to view Kepler’s works as he would have wanted them to be seen: not as dry formulaic articles on astronomy, but as the brilliant unveiling of a speculative thinker and scholar, believing in the harmony of the universe, to whom the formulae were not ends in themselves, but merely means and tools for the proof of his great harmonic formal ideas.
Here I will remark only that Kepler, in his Harmonice Mundi, attempted to find harmonies in the planetary distances. The idea that guided him in this was to coordinate the planetary orbits with the circumferences of the so-called “five regular solids,” to “box” them together so that they corresponded to the approximate distances. Kepler tried to realize this in his first work, Prodromus (later reissued by him with corrective notes as Mysterium Cosmographicum), and he approached it anew under a harmonic aspect in Harmonice Mundi, but no longer found the precision he was seeking, although the morphological value of the five-solid scheme as the background of the distances of the planetary orbits from the sun still seemed correct to him. He then set about analyzing the distances of the aphelions and perihelions of the individual planets harmonically amongst themselves. Regarding the extreme distances, i.e. the greatest and smallest distances from the sun (since the planetary orbits are ellipses), for the individual planets, Kepler found harmonic intervals only for Mars and Mercury; “however, if one compares the extreme distances of the various planets to each other, the first light of harmonics dawns” (Harmonice Mundi, Book V, Ch. 4). There, in close approximation, are the ratios
2/1 5/3 4/1 3/1 27/20 12/5 and 243/160.
But this did not satisfy him either. He then sought harmonies not in the distances, but in the velocities, in the daily heliocentric movements. Here, in the extremes of the individual planets (see Caspar’s translation, p. 301, as well as my Hörende Mensch, p. 177), he found harmonic ratios everywhere, with such close approximation that it could not be coincidental. A further investigation of these ratios, using our familiar operation of interval potentiation, then led him to discover his famous Third Law, which he used not as an end in itself, but as foundation for the proof of the existence of his heavenly harmonics. We know this from the fact that this law was given as the eighth of thirteen fundamental theorems (Book V, Ch. 3), which are “necessary in the observation of heavenly harmonies.” Furthermore, regarding this law, he writes (Mysterium Cosmographicum, tr. by Caspar, 2nd ed., Ch. 21, note 8, p. 137): “An explanation for the eccentricities has been found, not from the ideas just presented, but from the harmonies.” For the reader of this book, no further proof is necessary that for Kepler, the main issue was not the (modern) formula of his Third Law: f12 : f22 = r13 : r23 (“the squares of the orbital periods have the same ratio as the cubes of the major axes”) but the fact that this formula should be the exact proof for the harmonies existing de facto in the planetary movements and distances—harmonies that achieve their simplest sensory and psychical realization in music, which for Kepler is therefore an image of the heavens, i.e. of the planetary system. Why else would he devote pages in the conclusion of Book V to hundreds of these harmonic ratios, as the crowning of his work, with minute precision and love? And why else would he give, in the same chapter, all the laws of music, chords, melodies, major and minor, the scales, and finally the entire chord of creation in notes?
And all this, not before the harmonic derivation of his law, but after it, as proof of the great harmonic form-idea of the Harmonice Mundi, the “harmony of the world.”
The question of whether Kepler’s “harmonies” are still correct today can be answered absolutely in the affirmative. A comparison of Kepler’s aphelions and perihelions in regard to the distances from the sun with the current values shows, with the exception of Saturn’s aphelion, only very minor differences; the same is the case for the daily heliocentric movements, since these fluctuate around average values, despite the “perturbations” that have since become known. Uranus and Neptune also show Kepler’s harmonic ratios, and looking through the abovementioned work of W. Kaiser, one discovers so many further typical harmonic ratios that one will find Kepler’s “Harmonice” proven once again, under aspects of which Kepler himself knew nothing. And as for the “failure” of his polyhedron theory, i.e. the theoretically possible existence of only 6 planets through the existence of only 5 regular solids, this is only connected with the actual harmonic ratios insofar as he derived musical intervals from the polyhedron ratios in his Harmonice Mundi, and there found the theoretical anchor for the harmonies suspected to exist in the planetary system. These are present above all in the planetary movements; and Kepler would have been the first to abandon the polyhedron theory, and would have concentrated completely on harmonics, whose laws he would have proven for the other planets with equal delight and enthusiasm, as he did for the planets known to exist during his time. Kepler’s harmonics are still just as valid today as they ever were—for those with ears to hear them. Indeed, we can construct them much further and more broadly with current harmonic resources, especially in terms of the symbolic-spiritual side; the “P” system was unknown to Kepler.
With this, however, Kepler’s significance for modern harmonics is in no way exhausted. In his Harmonice Mundi, especially in Book IV, there is a series of fundamental and exceptionally important observations for the foundation of harmonic starting positions (if I may thus express myself); in any case, the entire work is saturated with a wealth of principial harmonic observations, examples, ektypic references, etc. We must fittingly and justly view Kepler’s Harmonice Mundi and A. von Thimus’s Harmonikale Symbolik as the two great harmonic cornerstones of modern times, the exhaustive study of which should be the first and foremost duty of every harmonist. In Kepler’s other works there is also a great deal of important material for harmonics, for example his commentary on the harmonics of Claudius Ptolemy, many letters to and from Kepler, and so forth.
Besides the passages cited in the text in my Hörende Mensch, Grundriß, and articles in the Schweizer Rundschau: the works of astronomer W. Kaiser, and those of the gestalt theorist Christian von Ehrenfels. Regarding Kepler: the great new critical collected edition (ed. M. Caspar et al., O. Beck-Verlag Munich, starting 1938), of which volumes 1, 2, 3, 4, and 6 have been released as of now (1944); and M. Caspar’s translation of the work on Mars (Neue Astronomie, ibid. 1929), Mysterium Cosmographicum (Das Weltgeheimnis, Munich, Oldenburg 1936), and Harmonice Mundi (Munich, Beck 1929). Moreover, Caspar’s Johannes Kepler in seinen Briefen (Munich, Oldenbourg 1930) and the Biographia Kepleriana (Munich, Beck 1936). Other writings on Kepler and others: Apelt, Reformation der Sternkunde (Jena 1852); L. Günther, Die Mechanik des Weltalls (Leipzig 1909); and the standard translation (with introduction and commentary) of Harmonice Mundi by W. Harburger (Johannes Keplers Kosmische Harmonie, Dombücherei, Inselverlag, Leipzig 1925), in which the “table of geometric-musical-astrological-astronomical correspondences” is noteworthy.
E. Britt’s booklet, Tonleitern und Sternskalen, tr. by Felix Weingartner (Leipzig 1927) is mentioned here only as a curiosity, and as an example of how one should not engage in such investigations. In harmonics we are used to somewhat more thoroughness and exactitude; above all one should never use the tempered system as a basis here, only pure-tonal values and ratios. Viktor Goldschmidt, in his Über Harmonie im Weltraum (Ostwalds Annalen der Naturphilosophie V, p. 51 ff.), investigates the planetary distances, among other things, by means of his “law of complication.”