Thursday, January 19, 2012

Mathemagical Solar System Song

Daniel Starr-Tambor’s Mandala (above) is a musical palindrome of 62 vigintillion (1063) notes, which represents musically all the rotational frequencies of the planets (and Pluto) in our solar system! (via Brain Pickings) I've always been a fan of the palindrome (which is something which remains the same if write forward or backward) since my name (Ele) is one of the shortest palindromes. Mandala boasts that it is the longest palindrome ever written, and I do not doubt it.

It's perhaps surprising how lovely the music is to hear, for such a mathematical approach to music composition - though the two languages, math and music, are quite naturally intimately entwined. He's not the first to think of the planetary motions in terms of music. He alludes to Bach (and Bach’s The Art of the Fugue) explicitly, with its contrapunctal mathematical and arguably Pythagorean structure.* The concept of the Music of the Spheres was quite a common way for scholars to think about the motions of heavenly bodies, up to the Renaissance and the Scientific Revolution. It can be traced back to the ancient Greek mathematical-mystic Pythagoras, who first linked musical pitch to the length of a vibrating string which produced it. Further harmonious sounds were produced by strings with simple (rational) length ratios. This fit well with his adoration of rational numbers.** Pythagoras, in his theory of the Harmony of the Spheres, proposed that celestial bodies (Sun, Moon, and the known planets, Mercury, Venus, Mars, Jupiter and Saturn) each emitted a hum based on their period of revolution. Recall, that it was assumed that the Sun, moon and planets orbited around the Earth. Further, that the ratios were harmonious, like those produced by strings of simple length ratios. Thus, for Pythagoras, music and astronomy were two sides of a single mathematical coin. Ptolemy's model of the geocentric solar system, wherein each of Sun, Moon, known planets, and stars (quintessence) were more-or-less pinned to a series of transparent, nested, rotating spheres encapsulating the Earth, dominated the Western world view for 1500 years. Thus "Music of the Spheres" referred to the 'harmonies' of the motions of these imaginary spheres.

The first real shift in thinking is of course attributed to Copernicus, who proposed a heliocentric model, where the Earth, like the planets, circled the Sun, and the Moon circled the Earth. This model was famously improved by Kepler, who, thanks to Tycho Brahe's immaculately recorded data, was able to show that planetary orbits are in fact elliptical (slightly more oval than a perfect circle), with the Sun at one of the two foci of the ellipse. He further showed that a line joining a planet and the Sun sweeps out equal areas during equal intervals of time. And, that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. These three facts are known as Kepler's Laws of Planetary Motion. You'll see that they refer to geometry and ratios, just like Pythagoras did before him. In fact, while we tend to honour Kepler as some sort of harbinger of modern science, his worldview was in many ways still Medieval. He did not think in terms of the simple, beautiful, elliptical solar system (as we still know it today). He was obsessed with proportions and attempted to explain astronomical (and worse, to the modern mind) astrological ratios in terms of music, in his own favorite publication Harmonices Mundi. His very first attempt to articulate his discoveries, now known as his three laws, was written in terms of musical notation (above). He wrote, "the movements of the heavens are nothing except a certain everlasting polyphony." (He ended up embroiled in a priority dispute with Robert Fludd, whose own harmonic theory is illustrated as the "THE MUNDANE MONOCHORD WITH ITS PROPORTIONS AND INTERVALS", the stringed instrument with planetary orbits, From Fludd's De Musica Mundana above). He also conceived of it in terms of polyhedra; he imagined the five Platonic solids (octahedron, icosahedron, dodecahedron, tetrahedron, cube) circumscribed by spheres, nested one within the other. This, he claimed, could explain the ratio of orbits of the six known planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). Kepler's insights, along with the concept of centrifugal force from Huygens, allowed Newton, Edmund Halley, and possibly Christopher Wren and Robert Hooke to deduce that gravitational attraction between the Sun and its planets decreased with the square of the distance between them. This in turn lead to Newton's Universal Law of Gravitation. So, we owe this groundbreaking idea as much to musical theory, and imaginary geometrical patterns, as rational, mathematically-based reasoning.

*For how Bach encoded his name into The Art of the Fugue, see one of my favorite books, Gödel, Escher, Bach by Douglas Hofstadter.

**We all recall Pythagoras for Pythagoras' theorem (the square of the hypotenus is equal to the sum of the squares of the other two sides for right angle triangles). Imagine his, and his cult's horror when they realized that for one of the simplest triangles, where "the other two sides" each have length of 1 unit, the hypotenus must have an irrational √2 length! The irrationality of √2 did not fit into the tidy rational, mathematical world of the Pythagoreans. It was perhaps the first of a long series of instances where the beauty of mathematics was mistaken for something tidy and controllable.

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