Book III of Kepler's Harmony of the World
In Book III we leave behind the domain of visual geometry. We'll again explore the principles of the organization of space—but this time as they are projected into the domain of our hearing: musical harmonics.
In this book, Kepler constructs the full harmonic system, in multiple octaves, and with multiple modalities. Just as in the first two books, he is not concerned so much with the sounds as sounds1, but with the principles of construction which lie behind what is sensed—with what is accessible to and found to be beautiful by the human mind.
“...and since it is Mind which shaped human intellects in such a way that they would delight in such an interval (which is the true definition of consonance and discordance) the differences between one and the other, and the causes of such intervals' being harmonious should also have a mental and intellectual essence, that is that the terms of the consonant intervals are properly knowable, but those of the dissonant intervals either cannot be properly known, or are unknowable. For if they are knowable, then they can enter the Mind and into the shaping of the archetype; but if they are unknowable then they have remained outside the Mind of the eternal Craftsman...”
—INTRODUCTION TO BOOK III
What you'll find, as Kepler did, is that the domain of harmonics, and beyond that, of music, the soul which animates it, opens up for us different aspect of these principles of organization than did our explorations in geometry. While Kepler was not a musical composer himself, it was certainly musical thoughts which inspired him, and indicated the potential of harmonics as a scientific language which could be used to understand the great composition of our Solar System.
Here, we present the construction of Kepler's full harmonic system (and its modalities,) the naming of the intervals, the impure harmonies which arise within the system, and how this necessitates a process of tempering (or adjustment of the intervals) in order to form a coherent, unified musical system. That process of tempering is something that will be encountered again in the final sections of Book V. In fact, the type of tempered harmonies that Kepler develops in astrophysics were far advanced of any of his contemporary musical theorists, resonating with what was to come with the genius of J.S. Bach, and, in both musical and celestial harmonies, betrays the presence of idea, of mind.
“Throughout we shall indeed speak of melody, that is harmonious intervals which are not abstract but realized in sound; yet to the educated ears of the mind the underlying reference throughout will be to the intervals abstracted from the sounds. For it is not only in sounds and in human melody that they yield their charm, but also in other things which are soundless, as we shall hear in the fourth and fifth Books.”
—INTRODUCTION TO BOOK III
1. Just as he did not study number and geometry “...in order to draw up accounts of merchandise, but to explain the causes of things.”