In the fifteenth and sixteenth centuries, musical theory was experiencing dramatic change and controversy no less than was astronomy. Conflicting views of harmony and the correct musical ratios fuelled the controversy. The question of which ratios produced consonances or dissonances made the argument a fundamentally mathematical one. Kepler, believing that the heavens incorporated the same created harmony that appealed to humans in fine musical chords, became fascinated by the problem. He set out to decipher the harmonies of the world by reconsidering musical theory in its entirety.
Kepler found himself favoring the system of 'just' intonation, documented by Gioseffo Zarlino in 1558. However, he referred to this system as Ptolemy's. Like almost all scholars of his time, Kepler was convinced that new knowledge would prove to be very old; and he became convinced that Ptolemy had proposed an almost identical theory to his own in his Harmonica. The major difference between the 'just' intonation system and the (reputedly Pythagorean) alternative was that "Ptolemy's" accepted thirds and sixths as consonances, whereas the Pythagorean system considered them dissonant. Since contemporary composers were utilizing consonant thirds and sixths in the production of polyphonic music, this became far more than a dry academic debate. One such composer, Orlando di Lasso, was repeatedly cited by Kepler in his Harmonices Mundi. Lasso's innovative compositions utilised chromatic techniques that Kepler believed in harmony with true musical theory - and hence able to affect the souls of listeners, raising them to joy or felling them into grief.
Kepler believed that "Ptolemaic" intonation derived from geometry - and hence from divine archetypes. The Pythagorean system was based on numbers alone, he claimed, repudiating the views of Robert Fludd. His own ratios were derived, not arbitrary. He claimed that geometry was the basis of all natural things.
Specifically, Kepler drew upon his work using the regular polygons to underwrite the Ptolemaic system. He used geometry to derive the ratios required for the musical theory he desired by referring them to the lengths of chords formed around circles by the inscription of plane shapes. In other words, while the planetary distances reflected three-dimensional geometry, the harmonies of music, astrology, and other phenomena reflected two-dimensional geometry.
Kepler's next task was to incorporate these consonances into his cosmology. He studied the path of each planet very carefully, attempting to find a relationship between the motions of the planets and Ptolemy's musical system. Finally, he discovered a property of the planets which fit the ratios startlingly well. Kepler found that the angular speeds of each planet at aphelion and perihelion, as measured from the Sun, produced consonant ratios. He was thence able to write down scales which he claimed each planet followed as a result of its changing angular velocity around its elliptical orbit. Mercury's range of notes would be the largest, since its eccentricity was the most marked; Venus's scale consisted of only one note. But these were not ordinary, playable scales. Kepler thought that each planet would produce a continually changing note, producing a tone similar to a siren. (Unfortunately, our reproduction does not yet replicate this effect).
Kepler found that the planets were able to form four harmonious chords. He proposed that one of these four chords was certainly formed at the time of Creation, and he suspected that another would mark the end of the universe. We do not know when in history he would have placed the other two.
|Click on any of the seven
scales to view the orbit and hear the music.|
Remember that all scales begin at the planets' aphelion position, rise to perihelion, and then go back down to aphelion. Also, all scales are displayed in 16th century German notation and are taken from the Harmonices Mundi.
|Mercury - begins at C and rises until E. This scale should be four octaves higher than the treble clef that would be associated with Saturn|
|Venus - both aphelion and perhelion positions are at E (this is not to say that there is no variation, just that there is not enough change to make the perihelion position an F). This scale is three octaves higher than the treble scale associated with Saturn.|
|Earth - begins at G and rises to A flat. This is two octaves higher than the treble associated with Saturn.|
|Mars - begins at F and rises to C. This is only one octave higher than the treble associated with Saturn.|
|Jupiter - begins at B and rises to D flat in the bass clef.|
|Saturn - begins at G and rises to B in the bass clef.|
|The Moon - begins at G and rises to C.|
The first two columns in the above chart display the musical ratios of the
extreme motions of neighboring planets. The first is when the aphelion velocity
of the higher planet is compared with the perihelion velocity of the lower.
The second is when the perihelion velocity of the higher planet is compared
with the aphelion velocity of the lower. The fourth column of the table gives
the 'observed' daily motion of each planet in minutes and seconds of arc. The
sixth column displays the musical ratios which fit most closely the ratios of
the extreme motions of the planets. The fifth column gives the daily arcs that
the planets would travel to give this musical ratio exactly. Kepler was struck
by the agreement between these numbers and the 'observed' values in column four.
Above is one of the two scales which Kepler constructed to illustrate the different
notes that the planets occupied at the aphelion and perihelion positions. This particular
scale begins with G (the aphelion position of Saturn) and is only missing the note A. It
lacks only the perihelion positions of Earth and Venus. Kepler called this scale cantus
durus, which loosely translates as 'sharp' or 'hard' scale. Our own concepts of major
and minor had not yet been agreed upon.
Above is the other scale. Its lowest note is once again G, corresponding to the aphelion position of Saturn; it is missing only the note F. This scale lacks the aphelion position of Saturn, the aphelion position of Venus, and the perihelion position of Mars.