Musical Spheres

Fellow classmate and blogger The Mad Physi gave me a beautiful gift this quarter: a book called QUADRIVIUM.

This book is actually a collection of four ancient books, partly written during the time of Pythagorus, and have been studied by Cassiodonus, Philolaus, Timaeus, Archytus, Plato, Aristotle, Eudemus, Euclid, Cicero, Philo the Jew, Nichomachus, St. Clement of Alexandria, St. Origen, Plotinus, Dionysius the Areopagite, Bede, Alcuin, Al-Khwarizmi, Al-Kindi, Eriugena, Gerbert d'Aurillac, the Bretheren of Purity, Fulbert, Ibn Stina (Avicenna), Hugo of St. Victor, Bernardus Silvestris, Bernard of Clairvaux, Hildegard of Bingen, Alanus ab Insulis, Joachim of Fiore, Ibn Arabi, Grosseteste ("the great English scientist"), Roger Bacon, Thomas Aquinas, Dante, and Kepler. (Talk about name dropping.)

In Book IV, harmonics, scales, chord progression and more are discussed in the most fascinating way, with amazing visual representations of harmony through the use of a harmonograph (speaking of which, I am going to build one, and you can, too!) There is a section in this book entitled The Music of the Spheres, and the spheres it discusses are the planets of our solar system. I've heard of a more general concept like this before, more of a symphony of the entire universe, and I would like to share what the book had to say about "planets playing in tune" and some more recent developments on the subject...

As you may know, Kepler studied the motion of the planets. During his study, he wrote Harmoniae Mundi, or "harmony of the world," in which he compared the planets' angular velocities with harmonics. He set about calculating these harmonies based on the association previously given the seven known "planets" (including Earth's moon) with the seven musical notes. Below is a photo from the book showing the comparison between the ancient system and Kepler's interpretation.

The pentagons inscribed within the
(at the time this book was written) known orbits of Mercury and Venus and by Earth and Mars (pictured at right) are used in the book to prove the harmony of the worlds. Book II is all about geometry, and it was highly regarded during the time these books were written and studied; therefore, any explanation using geometry was not only a good argument, but it was actually regarded as being sacred.

The illustration below depicts two models which show how the planets were believed to orbit and sadly reveal a source of discord in the ancient sheet music...

Heavenly Harmony and Earthly Harmonics, a paper from the Quarterly Journal of the Royal Astronomical Society, tells us that the ancient Greeks, including Plato and Aristotle, believed in the "harmony of the spheres," and that the planets played music as they swept through space. Throughout the Middle Ages, this belief persisted, with the romanticized piety of notion that this music was the planets themselves praising God. In 1596, Kepler wrote Mysterium Cosmographicum, within a couple years after Shakespeare wrote about the "harmony of heaven" in The Merchant of Venice. In Mysterium, Kepler used simple geometry to provide a logical explanation for this accepted harmony. I assume this is the same geometry as I showed you earlier with the pentagons. However, although most of his calculations were astonishingly accurate, they were based on the assumptions that there were only six planets, as you saw in the inaccurate illustration of the motion of the planets.

Later work by less famous Molchanov cast Jupiter as the "conductor" of this orchestra, and included the entire solar system. His "commensurability" concept which uses linear equations of the planets' orbits and their frequencies to explain their resonance is much more accurate, and much more rigorous.   Here are the linear equations of the frequencies, where m denotes frequency:

In every case, he is off in his calculations by less than a percent, and his second equation gets replaced by 

.

Also, not only do these equations work, there are also accurate equations for the planets' moons. Computer systems are currently being programmed and tested in order to better understand these resonances. These models have not found fault with the numerology of Molchanov, meaning that there really is a harmony to the solar system. The frequency ratios of arbitrary solar systems in these computer models 2, 9/4 to 7/3, and 5/2 are favored, and these are representative of our solar system. 

However, there is also discord predicted in the symphony of all solar systems. Just as in a poorly engineered bridge, improper resonances lead to the destruction of weak celestial instruments: inadequately small masses and or playing out of "key" (with the wrong frequencies), according to the computer models. Smaller masses with unstable orbits will either be "swallowed up or smashed." (I imagine the cacophony of someone banging, disharmoniously, on the low keys of a piano at this moment.) Hopefully, this already happened in our solar system, or it won't affect us while we are living in this beautiful evolution of harmony.

If you found this interesting, be sure to check out what my fellow classmate wrote about galactic geometry on her blog, The Mad Physi, and read what my other classmate wrote about the physics behind music in his blog, Up and Atom! Also, here is a video of a harmonograph in action!