The Ptolemaic Evolution
The Greeks struggled to advance their understanding, but it was to no avail until one of the most famous astronomers and philosophers of all time came onto the scene. Ptolemy, Greco-Egyptian by descent, performed that majority of his work in the Egyptian intellectual mecca of Alexandria.
Ptolemy’s model for the universe came on the back of several advances that were based partially on observation and partly on mystical pleasures of symmetry. Although he was an advocate of observation-based analysis, the brute fact is that metaphysics and physics had not yet been fully teased apart. The familiar Earth-centric model seen in most texts from the 4th. century B.C.E. onward was developed initially by a little-known astronomer named Eudoxus, and a well-known one, Hipparchus. The only reason most people haven’t heard of Eudoxus is because Aristotle and his quasi-scientific method was the dominant force for 1000 years, and Aristotle's model was also considered a high water mark of the Greeks.
Eudoxus' model placed the Earth at the center of the universe, and each heavenly body rotated in a perfect circle around it. Aristotle included the addition of stationary spheres, with a sort of lubricating sphere separating each moving one so the theoretical motion of one sphere wouldn’t influence any others. He then added some other babble to make it all pleasing to the philosophical mind—a failing of process for which posterity, and for that matter Ptolemy to some degree, would look upon him with the same wistful awe owed to the likes of Odysseus for defeating a non-existent cyclops.
What is somewhat shocking is that Aristotle overlooked problems in his model that were obvious to his very own naked eye. First, depending on the distance from the Earth in the model we know today, the planets appear brighter and dimmer depending on how close they are to us. Also, how the outer planets occasionally moved in retrograde motion may as well have been caused by Aquinas' angels flapping their wings. Likewise, they also seemed to speed up and slow down from time to time. Most damning, it didn’t allow any predictions to be made from the model—it was simply a statement of how they believed the universe was constructed. Hipparchus would come up with an initial solution, and Ptolemy would develop it until it mostly provided the right answers for the wrong reasons.
To address retrograde motion as well as the relative brightness, it required that, in addition to the movement of the planets on circular orbits called a deferent, each planet also had an epicycle—an orbit around the point on the deferent. Hipparchus was the first to suggest epicycles, but Ptolemy improved the math and scope significantly. It is difficult to conceive in words but simple on paper:
The initial calculations handled each planet individually, but in later work Ptolemy was able to integrate them into a coherent model in which all the planets' motions worked individually and in concert. The pleasing aspect of his work is that it indicated distinct ratios for each planet’s distance from the Earth. The moon, for example, was 1:48 Earth-radii away. This is not very close to the modern value, but the worst was the outer shell of the stars: 1:20,000. Modern scientists believe this valuation to be...slightly off. Here is Ptolemy’s Earth–planet ratio chart:
In fact, none of these ratios is close to the truth, but this was about the best anyone could do until sufficient progress was made in the field Galileo ennobled: optics. For the entirety of human history until his revolution, Earth observation would be the only means of modeling the universe. Mostly reliable predictability would have to suffice for truth.
Of course, “mostly reliable” is the key term. The orbits of Mars and Saturn worked out provided you were willing to add a few degrees here or there for as good a reason as assuming angels were pushing Mars backwards and forwards in the sky. I suppose something can be said for at least narrowing their flight path. Also, when he realized the model didn't accurately account for the changes in speeds, in desperation he fudged the math and added the equant—a point opposite the Earth similar in concept to Philolaus' Counter-Earth. The equant and the Earth were both offset from the "center" of the universe, with the planets traveling over equal distances relative to the equant.
Points like this would cause a few of the great Islamic astronomers to hold Ptolemy in outright contempt, not necessarily because he was wrong—progress depends on the incorrect predictions of one’s predecessors—but because it was patently obvious that Ptolemy knew it. In some ways you can hardly blame the man. He'd spent his entire life dedicated to the Earth-centric model build around the philosophical perfection of the Greeks, and at some point a human with a finite lifespan can't be expected to just dump their life's work and start from scratch. History seems willing to credit him for an earnest and profound attempt, and this seems fair.
The very first sentence prefacing Ptolemy's magnum opus Almagest reads, “Those who have been true philosophers, Syrus, seems to me to have very wisely separated the theoretical part of philosophy from the practical. For even if it happens the practical turns out to be theoretical prior to its being practical, nevertheless a great difference would be found in them.” This is, however, followed up in the proper body of the first book by a lengthy explanation of why the Earth must be the unmoving center of the universe—an explanation at odds with the data of “practical philosophy”. In every way, a summation of a life's work.
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The great example that Ptolemy set, and a consistent pattern of great thinkers of the past, is one that has largely been abandoned by civilization today. He was a polymath, and he spent his life exploring a vast and diverse collection of mysteries and sciences from astronomy to optics to—thank you very much—music. It is only until comparatively recently that Rodin's Thinker stood up from his block and grabbed either a violin or a test tube, and it's a shame in many ways. Polymaths would dominate the fields of music and astronomy all the way up to the man who discovered Uranus. Therefore, we now turn to Ptolemy's valuable contributions to the mysteries of music.
This perfection of ratios and an elevation of Greek cosmological principles bled over into Ptolemy’s attempt to correct the Pythagorean Dilemma. After all, if the harmoniousness of the celestial spheres could be unlocked then there must be an equally elegant musical solution for the music they produced. Like his cosmological model, his technique was to apply the closest perfection of whole number ratios he could, placing the practical above the theoretical ideals that had limited the Greeks. This system is known as “just intonation”, and for interrelated reasons with the cosmos would serve musicians almost exactly as long as Ptolemy’s model would serve astronomers.
Just intonation solved the problem of Pythagorean harmony by discarding the need for the pitches to be generated organically our of a single pitch by extrapolation. Instead, he focussed on finding the lowest possible ratios for each note, and therefore the least possible dissonance. He came up with the following table:
The advantage to this system is that it creates the least possible interference within a particular scale. Even better, it makes tonal harmony pleasing, in the sense that a major triad sounds excellent. In fact, even to modern ears, a chord built upon notes with these frequencies sounds more “in tune” than the system we use today. In fact, it is more in tune, at least in a certain sense. What Ptolemy had created was an idealized scale, not an idealized harmonic continuum. If you start on A, the ratio of A–B is a perfect 9:8. If you tune the notes as above relative to A and then try to play a scale starting on B, well, just do the math.
The ratio of A–B is 9:8, but the ratio of B–C#, the beginning of a B scale, is 10:9, and it only gets worse. Just like the cosmological model, it worked only insofar as you were willing to tolerate or correct for a few obvious flaws. That being said, it is charming to note that the octave ratio is 2:1, and the ratio of the Earth to the sphere of the stars was 20,000:1.
Just as his heavenly bodies model lacked a certain perfection, I’m sure it was more than slightly annoying that this didn’t perfect the musical universe any more than the actual universe. The one maggot that probably did bore its way all the way to the center of his brain was that there was one more heavenly sphere than musical notes in a scale, so not even the whole concept of the harmony of the spheres would be completely coherent. However, if one is willing to dispense with a detail here or there in one realm, one may as well get comfortable with others.
In the end, both the cosmological and harmonic systems were accepted for the same reason: from the mathematical and scientific ground upon which they stood, they provided the least interference. Besides, the Church loved it. Outside the stars there was plenty of room for an infinite god who could dip his fingers into our little fishbowl any time he wanted to tickle St. Theresa to orgasm and allow the Crusaders to wade knee-deep through Muslim blood in the Temple of Jerusalem, and just like the Bible said, we were the focal point of all creation. Scientists on the other hand aren’t willing to put flawed theories in a glass case so the collection plate can be happily filled every Sunday. Despite threatening to burn Copernicus at the stake for suggesting the Sun was the center of the universe, the progress of mathematics that allowed music to evolve into its current form would this time inform the science necessary for explaining Ptolemy’s cosmic problems.
Both would occur within a few years of each other because, serendipitously, two very famous historical figures also happened to be father and son.