DAILY DOSE OF BEETHOVEN (June 30, 2020)
Today, we will delve deeper into Johannes Kepler’s role in the development of the Well-tempered System. We will discuss Kepler's discoveries in Astronomy and music, which innovated both fields of studies.
Johannes Kepler (December 1571 – November 1630), as an astronomer, is universally recognized for his discoveries in astronomy, including the discovery of universal gravitation. Yet, his equally rigorous discoveries in music, which changed our understanding of the very nature of the musical system, are dismissed as mystical. Kepler, in 1619, published one of the most important scientific treatises to date, which is titled “The Harmony of the World.” For context, the Thirty Years War began in 1618, which is at the end of over 100 years of religious war that destroyed Europe, which only ended in 1648 with the Treaty of Westphalia. At the time, the earth is still viewed as the center of our solar system (geocentric). It was only in the year 1623 that Galileo published his “Dialogue Concerning the Two Chief World Systems”, which discussed heliocentricity among three people: one who supports Copernicus' heliocentric theory of the universe, one who argues against it, and one who is impartial. This caused Galileo to be interrogated by the inquisition and forced to recant his heliocentric theory. It is in this political atmosphere that Kepler, whose own mother was burned as a witch, published his discoveries.
KEPLER'S STRONG HYPOTHESIS
Kepler wrote that he KNEW that the musical ratios would be found in the organization of the solar system and spent 25 years looking for it, including in all the wrong places, such as the masses of the planets and his earlier discovery of the planets' ordering according to the "platonic solids." (See diagram). How could he be so sure? What was his evidence?
In Book Three of “Harmonice Mundi” (The Harmonies of the World), Kepler said that he intended to overthrow the axiomatic assumptions held for 2,000 years that the cause of the intervals lies in rational numbers, such as 3/2 for the fifth. This assumption had held music back, because those ratios almost, but didn’t exactly work. They have to be tempered. Kepler says this of the cause of the intervals:
“Since the terms of the consonant intervals are continuous quantities, the causes which set them apart from the discords must also be sought for among the family of continuous quantities, not among abstract numbers, that is, in discrete numbers, since it is MIND which shaped human intellects in such a way that they would delight in such an interval..”
WHERE HE FOUND IT!
In Book Five, Kepler took his life-long study of the solar system, and found the musical intervals with amazing accuracy, in the ratios of a planetary orbit, between the angular velocities at the perihelion and aphelion. For simplification, look at the animated orbit of a planet provided (https://giphy.com/gifs/MYCbbIYhv6MO3wE1CT). Notice that it moves faster when closer to the sun (perihelion), and slower when farther away (aphelion).
But wait! That is also one of Kepler's great breakthroughs in astronomy! His second law states that the orbit of a planet will sweep out equal areas in equal times. (see the diagram of an elliptical orbit). Before Kepler, a false axiomatic assumption prevailed. It assumed that the orbits of the planets must correspond to perfect circles. This in a way, is similar to the false axiomatic assumption that musical intervals must correspond exactly to rational numbers (fractions).
As long as the assumption of circular orbits persisted, even Copernicus' (1473 –1543) heliocentric model oblige the planets to perform loop-de-loops known as epicycles, in order to fit observations (see comparison of the Ptolemaic geocentric model, and the Copernican heliocentric model. Both require epicycles.) The theory, and the physics, did not match. Kepler's discovery of the actual elliptical nature of the orbits brought theory and practice into coherence. Without that elliptical nature, the orbits would not have different angular velocities, and there would be no musical ratios.
SCALES
Kepler found the ratios of individual intervals in each planet. When he compared the ratios between planets, he found two scales, corresponding closely to our modern major and minor scales.
TEMPERING
What about the other false axiom: rational numbers alone as determining the harmonic values of intervals? In Book Three, Kepler criticized those who tried to base music on mathematics.
“The Pythagoreans were so much given over to this method of philosophizing by numbers, that they did not even stand by the judgment of their ears…they marked out what was melodic and unmelodic, and what was consonant and what was dissonant from their numbers alone, doing violence to the natural prompting of hearing. “
In another word, human voices cannot distinguish these odd intervals, he says, but:
“It is possible for strings to be tuned that way, since they are inanimate and do not impose their own judgment, but follow the hand of the foolish theorist without the least resistance.”
Even though the ratios Kepler found in the planetary orbits correspond very closely to those numerical values, in Book Five, he sets up a table showing how the ratio of the angular velocity at the perihelion and aphelion of a planet, varies slightly over the course of a year. They are not fixed "perfect" ratios. Thus the possibility of tempering, of making the slight changes necessary is something the musical system can adopt from the solar system. Kepler's notion of tempering is not a practical adjustment, but again, has its origin in Mind. From Book Three:
“Friendships are given life by harmonic tempering. For what concord is to proportion, that love, which is the foundation of friendship, is to the whole compass of human life…Although friendship cannot survive frequent injustices, yet it rejects laws (strict rules), and refers everything to the sound and sober judgment of love, dispensing now equally, now proportionally, and when neither, always dispensing what seems to be in the immediate situation to make for the preservation of love, which is also goaded on, as harmony is by discords…by a few injustices, and receives its strength by free forgiveness of them.”
The Well-Tempered system is not a set of fixed ratios. Tempering is a human activity, in industry as well as music, that calls for the human mind to bring nature into accord with itself!
Kepler was known to many musicians who preceded Bach. Andreas Werckmeister, in his “Hypomnemata Musica”, a work calling to establish a Well-Tempered system, wrote:
“I wish to say something more about the excellent imperial mathematician, Johannes Kepler’s opinion, which he stated in the five volumes of Harmonice Mundi…that the Archetype is the cause in that it not only moves the stars to a more beautiful harmony (indeed not sensuous) but even more that it prepares the minds and souls of men so that they exist in such harmony and are moved through the heavenly motions, and are made joyful through sensuous, earthly music.”
Kepler overthrew the false axioms that held back both astronomy and music, and put both into harmony with the laws of the universe and its Creation. He is absolutely unique.