Musica Mundana: The History of Musical Astronomy. The idea of celestial harmony, or musica mundana (music of the worlds) as it

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1 Maxwell Phillips May 4, 2015 Musica Mundana: The History of Musical Astronomy The idea of celestial harmony, or musica mundana (music of the worlds) as it was called in the middle ages, has a history at least as old as Pythagoras and continues to play a part in contemporary astronomy in the theory of orbital resonances. The idea that celestial bodies move in relationships to each other that might by described harmonically, although stated and interpreted many ways, has been a constant presence in astronomical thought, appearing at important times in the field s history as an indicator of its metaphysical goals and assumptions. Comparing these different formulations throughout history, we can achieve a glimpse into the evolution of the field itself and its relation to the enterprise of human knowledge generally. In the ancient world, astronomy was considered part of the quadrivium of liberal arts, not a natural science such as physics. When, during the scientific revolution, thinkers such as Johannes Kepler laid the groundwork for the physical science of astronomy, Kepler used the language of music to describe his findings. This language stays with us today, for example in the description of simple harmonic relationships between the orbital periods of satellites as resonances. One of the earliest extant accounts of a theory of celestial harmony is found in Pliny the Elder s Natural History. This work, written in 77 AD two years before its author s tragic death on a rescue expedition during the eruption of Mt. Vesuvius, contains a description of the musical thought of Pythagoras, a Greek philosopher writing half a millennium earlier. Pythagoras described astronomical relationships

2 in musical terms: But Pythagoras at the same time uses the Terms of Music, by calling the space between the Earth and the Moon a Tone; saying that from her to Mercury is Half a Tone: and from him to Venus about the same space. 1 The distance between different planets is described as a musical interval, a step on a musical scale. To Pythagoras this was more than just a metaphor; different planets moved according to different musical modes: In this, Saturn moveth by the Doric Tune; Mercury by Phthongus, Jupiter by the Phrygian, and the Rest likewise. 2 The association of the physical movements of the planets with musical modes suggests a connection between the planets and human life on earth; different musical modes, we learn in Aristotle s Politics, were thought by the Greeks to induce different responses in humans. The Dorian (corresponding in Pythagoras to Saturn) for example was said to calm and stabilize the mind. Pythagoras s astronomy might thus be described as an attempt to provide a musical-physical basis for astrological beliefs common in the Greek world. The importance of a harmonic view of the cosmos and its relationship to music for an ancient understanding of life on Earth comes into clear focus in Boethius, a Christian philosopher active in early 6 th century Italy. Boethius s seminal De institutione musica was an important early text for both European music theory and astronomy. Following Aristotle, this text posits music as we understand it today and astronomy as two different aspects of the same phenomenon: the musical nature of the universe itself. The music made by instruments and the human voice is 1 Pliny, and Philemon Holland. Pliny's Natural History in Thirty-seven Books. London: Barclay, Ibid. 59.

3 compared to the music of the internal workings of the human body, musica humana, and the music of the cosmos: musica mundana. This musicality of the celestial bodies was critical to the medieval Christian understanding of God s perfect creation. This state of perfect concord, which the Greeks called harmonia, was the musical nature of God s universe. The music of the celestial spheres was not a metaphor but the harmony of creation itself, with profound effects on everyday human life. The mathematical model underpinning these theories was also derived from Pythagoras, who discovered the relationship between the length of a string and the pitch it produces. By dividing a string into integral numbers of equal segments Pythagoras was able to produce a series of perfect intervals, simultaneously inventing the first scientific tuning system and the mathematical harmonic series. The influence of this logic can still be seen in western astronomy in Bode s Law, a late 18 th century attempt to describe the distances between planets and the sun according to a geometric series. This series, successful in predicting planetary orbits until Neptune, represents a similar harmonic view of the cosmos. The difference between Bode s approach and Boethius is primarily one of procedure: while ancient thinkers concluded the order of cosmic harmony from abstract ideas of mathematical and musical perfection, Bode sought mathematical explanations for phenomena he measured and observed. These explanations were used to make predictions, which could be measured, culminating in Bode s triumphant discovery of Uranus in the predicted position. Whereas the ancient thinkers believed they were describing the universe s hidden nature, Bode attempted to measure the solar

4 system and find a mathematical explanation for those measurements. In more recently astronomy, although Neptune s position significantly closer to the sun than the predicted distance seemed to discredit Bode s Law, the discovery of other planetary systems and lunar systems that display similar regular planet spacing has caused some astronomers to reevaluate possible explanations for the great predictive success of Bode s law for most of the planets in our solar system. The transition to a more empirical, evidenced based approach for astronomy coincided with a similar movement in music theory. As Kepler and Galileo worked to challenge the Aristotelian view of a perfect cosmos, Gioseffo Zarlino and his student Vincenzo Galilei, the father of Galileo Galilei, challenged the Pythagorean vision of a perfect tuning system using empirically obtained evidence. They argued that a perfect tuning system was not possible, and that the use of perfectly tuned fifths to generate all intervals, as in the Pythagorean system, would result in imperfections in other intervals, calling the difference between theoretically perfect tuning and the practical result a comma. 3 This result was revolutionary; it implied that it was impossible to have a perfect tuning system. Tuning some intervals according to the harmonic ratios described by Pythagoras (2:3, for example, for a perfect fifth) necessarily leads to other intervals being imperfect. As music theory and astronomy were so closely related, compared to Aristotelian and Pythagorean doctrines about the harmonious nature of the universe this must have seemed like a claim that the universe was essentially imperfect. In musical 3 "Gioseffo Zarlino", in The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie. 20 vol. London, Macmillan Publishers Ltd., 1980.

5 terms, the universe was not perfectly consonant. This early idea of musical incommensurability implied its celestial counterpart. To the north, a German astronomer named Johannes Kepler was making a similar claim. Kepler s measurements of the parahelia and aphelia of the different planets did not conform to a perfectly harmonic model as predicted by the ancient theorists: It was found that all the extreme movements of the planets had not been adjusted perfectly to one natural system or musical scale, and that all those which had been adjusted to a system of the same tuning did not distinguish the pitches [loca] of that system in a natural way or effect a purely natural succession of concordant intervals. 4 Although Kepler himself believed that the perfect consonance of all the planets was possible, if extremely uncommon, he thought had perhaps occurred only once, at the beginning of creation. Kepler understood this as evidence of creation itself: But if only one sextuple harmony can occur, or only one notable one among many, indubitably that could be taken as a sign of the Creation. 5 What is revolutionary here is that Kepler drew his conclusion about the current incommensurable state of the solar system from measurable evidence, rather than attempting to derive it from theo-philosophical ideas. Kepler s first major astronomical treatise, the Astronomia Nova, describes itself on its title page as a physica coelestis, a celestial physics. While today the close relationship between astronomy, mathematics, and physics is fundamental to our understanding of these fields, in Kepler s time astronomy and physics had little 4 Kepler, Johannes, and Charles Glenn. Wallis. The Harmonies of the World. Charleston, SC: BiblioBazaar, Print Ibid. 296.

6 to do with each other. The evidence-based approach Kepler took was an important part of the shift towards a more scientific astronomy, and Kepler s work served as the basis for Newton s physical description of the cosmos in his Principia, which finally established astronomy as a branch of physics by deriving Kepler s Laws from the Law of Universal Gravitation. However throughout his work, as we have seen, Kepler still envisioned astronomy as a musical undertaking. In the Harmonices Mundi, Kepler goes so far as to use musical notation to describe the relationships between the planets (see image on right 6 ). In this image we see the harmonies Kepler was able to derive from different positions of the planets in their orbits. Mercury, which has a highly eccentric orbit, gives Kepler three distinct pitches at different positions. These consonances are described by Kepler as approximate, not perfect, but even their error is described in the musical terms of comma and diesis, the same terms invented at the time to describe inaccuracies in the Pythagorean tuning system. 7 The influence and importance of a musical approach to astronomy is unmissable. Kepler s third law, which related the period of a planet to its distance from the sun, and which is laid out at the end of the Harmonices Mundi, was described as the harmonic law. 6 Kepler Ibid. 288.

7 Modern astrophysics retains much of the musical basis of Kepler s thought, although it understands it very differently. Today the language of acoustics permeates astronomy in descriptions of resonances between orbiting objects, including the famous relationships between the orbital periods of Neptune and Pluto. As an example, I will discuss the 1:1 commensurability of Earth and the Asteroid Cruithne 3753 to demonstrate differences between contemporary conceptions of celestial harmony and older thinkers. Cruithne exhibits a 1:1 mean motion resonance with Earth, meaning that the average period of Cruithne s orbit about the sun is exactly one year. While on the surface this description seems extremely simple, allowing not even for a Keplerian comma of error, our knowledge of this asteroid s orbit is in fact very complex, accounting for influences from the Sun, the Earth, Jupiter, and to a smaller extent Saturn. 8 One might imagine an asteroid in an orbit very similar to Earths, though at a different phase in any point in time, like two horses on a carousel. In actuality, Cruithne has a highly eccentric orbit: one of the few know examples of the horseshoe type. The asteroid is only very rarely orbiting the sun with the same speed as the Earth (in discussions of resonance, the term speed is usually employed not to mean real velocity or even angular velocity, but instead frequency, the inverse of orbital period), but is half the time orbiting the sun faster than earth, appearing to be catching up to our orbit, the other half slower, falling behind. From the perspective of the Earth, the asteroid appears to be following a horseshoe-like trajectory, wherein the Earth occupies the position of the horseshoe s opening. Thus if the asteroid begins 8 Wiegert, Paul A., Kimmo A. Innanen, and Seppo Mikkola. "The Orbital Evolution of Near-Earth Asteroid 3753." The Astronomical Journal (1998): Web.

8 catching up to earth, when it comes close enough the gravitational effect of the earth on the asteroid will reverse its direction relative to Earth by slowing its orbit, until the to opposite effect occurs at the other end of the horseshoe. The horseshoe shape itself describes the compound effect of many smaller orbits, which appear beanshaped from the perspective of the earth. This effect is called shepherding. One full cycle is completed in approximately 770 years. 9 While we recognize the traditional impulse to describe this sort of relationship as resonant, several factors distinguish it from traditional ideas about the music of the spheres. While ancient astronomers, particularly Aristotle and Ptolemy, assumed that the heavens were immutable, Cruithne demonstrates instability a very short timescale. Whereas as late as Kepler astronomers assumed that the present arrangement of the sun s satellites was as old as time itself, our present description of Cruithne offers a most less stable picture. Predictions of future gravitational interactions with Mars and Venus suggest that it could be ejected from its orbit, and possibly the solar system, in as early as 5,000 years, and it is believed that the current orbit of the asteroid could be as young as only 500 years (an extreme estimate). On an even shorter time scale, the Lyapunov time of the asteroid is estimated to be around 150 years. 10 (The Lyapunov time is the time frame on which a chaotic system s development can be reasonably predicted, with prediction becoming exponentially less accurate with proceeding time.) This means that we are not able to accurately predict what path Cruithne will take only 150 years in the future. The opposite of a harmonious and fixed solar system, this 9 Wiegert et al. 10 Ibid.

9 scientific model is a description of a chaotic and unstable relationship that is irregular and temporary. This description of chaotic interaction contradicts the older ideal of natural law, wherein all of God s creation should have a discoverable rule as its basis. In astronomy, the movement away from a musical picture of the cosmos accompanies this shift. Resonance phenomena today are sometimes understood in themselves as instabilities, contrary to the traditional musical idea of the solar system. One such example is the Kirkwood gaps in the asteroid belt between Mars and Jupiter. This asteroid belt had several gaps, areas left empty by gravitational interactions with Jupiter. Prominent gaps include the 2:1, 3:1 and 5:3 resonances. As before, the resonance ratio refers to orbital period, so an asteroid in a 5:3 resonance with Jupiter would orbit the sun 5 times in the same time Jupiter did three times. Because of Kepler s third law, which relates a satellite s period to its aphelion, such a resonance can be extrapolated to a distance ratio from the sun. At the distances corresponding to these resonances, the asteroid belt is strangely empty. Instead of providing stability, these resonances increase instability, eventually ejecting whatever asteroids were present from their orbits. The few asteroids present in the gaps today, for example the Alinda family (in 3:1 resonance with Jupiter) are believed to be experiencing interactions with Jupiter that will gradually exaggerate their eccentricities, bringing them close enough to a terrestrial inner planet, likely

10 Mars or Earth, to expel them from their current orbits through gravitational interaction. 11 Although I have intentionally chosen examples of resonance effects that radically depart from the ancient expectations of harmonious stability that their name and history might suggest, other descriptions of resonance in the solar system seem more musical. One example is the well-known 3:2 resonance of Neptune and Pluto. This resonance, described as perfect because it is neither approximate nor coincidental, is held precisely (on average) by gravitational interaction between the two planets, taking into account the resonance of the librations of their respective perihelia. These resonances are stable over an incredibly long period of time 12. If the solar system were, as Kepler might have imagined, a giant harp, Pluto and Neptune would be strumming a constant, absurdly low-pitched, and slightly vibrating fifth, tuned perfectly according to the Pythagorean scale. From the beginnings of western astronomy to the present day, the field has been influenced and inflected by its relationship to music. Musical thought and theory, especially theories of harmony and consonance, have shaped the way astronomers conceive, understand and express their theories. From astronomy s twin place with music in the Greek Quadrivium of the liberal arts through its expression of musica mundana, the perfect harmony of the cosmos, through its discovery of incommensurability and Kepler s search for musical meaning within the dissonance he observed, and in theories of orbital resonance today, musical 11 Kirkwood, D. The Asteroids, or Minor Planets Between Mars and Jupiter. Philadelphia: J. B. Lippencott, Renu Malhotra (1997). "Pluto's Orbit". Retrieved

11 thought has appeared in astronomy at various times as the condition and expression of the assumptions, goals, and self-understanding of the field throughout history. Works Cited "Gioseffo Zarlino", in The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie. 20 vol. London, Macmillan Publishers Ltd., Kepler, Johannes, and Charles Glenn. Wallis. The Harmonies of the World. Charleston, SC: BiblioBazaar, Print Kirkwood, D. The Asteroids, or Minor Planets Between Mars and Jupiter. Philadelphia: J. B. Lippencott, Pliny, and Philemon Holland. Pliny's Natural History in Thirty-seven Books. London: Barclay, Renu Malhotra (1997). "Pluto's Orbit". Retrieved Wiegert, Paul A., Kimmo A. Innanen, and Seppo Mikkola. "The Orbital Evolution of Near-Earth Asteroid 3753." The Astronomical Journal (1998): Web.

Resonance In the Solar System

Resonance In the Solar System Steve Bache UNC Wilmington Dept. of Physics and Physical Oceanography Advisor : Dr. Russ Herman Spring 2012 Goal numerically investigate the dynamics of the asteroid belt