The Music of Pythagoras (57 page)

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Scholars such as Kahn think these men were not fictional and that their words reflected a much older line of Pythagorean speculation.

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In search of the source of Iamblichus’ lists of Pythagoreans, Burkert believed he had narrowed down the possibilties, conclusively, to Aristoxenus (Burkert, p. 105, n. 406).

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When someone asked what the practical use of one theorem was, Euclid turned aside to his slave, sniffed, and muttered, “He wants to profit from learning, give him a penny.” The Pythagorean aphorism was “A diagram and a step (an advance in knowledge), not a diagram and penny.”

^†
The three surviving books in which he included material about the Pythagoreans are
Metaphysics, Physics
, and
On the Heavens
.

^†
Recall that the regular solids each fit neatly into a sphere, and the fifth is close to being a sphere.

^†
Copernicus would point to Heracleides Ponticus as an ancient precedent when he presented his hypotheses in the sixteenth century. The Earth also rotated in Plato’s
Timaeus
, and the idea was probably not original with either man, for Philolaus and possibly earlier Pythagoreans thought part of the apparent movement of the heavens was caused by the movements of the earth. Copernicus also referred to Hicetas and Ecphantus of Syracuse.

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The
Elements
was translated by Boethius in about
A.D
. 480, but not until
A.D
. 1120, when Athelhard of Bath translated it again, this time from Arabic into Latin, did mathematicians begin to appreciate its worth.

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Cicero’s life, and his political life, began when Rome was a republic and ended after the assassination of Julius Caesar and the beginning of the reign of Octavian (Caesar Augustus). He was a strong supporter and defender of the republic and strove on its behalf during the civil wars.

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The Romans continued to use this formation effectively through the years of their republic and in the expansion of their Empire.

^†
Alcibiades’ reputation for lack of discipline and unscrupulousness was later used to support the charges brought against Socrates of corrupting the youth of Athens, which resulted in Socrates’ death sentence.

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Pliny lost his life when his insatiable curiosity about natural phenomena tempted him too close to the erupting Vesuvius.

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Cicero made several references to this celestial phenomenon that had appeared in the year 129
B.C
. The scientific name is parhelion, in the vernacular a mock sun or sun dog. The appearance is of two extra suns, one on each side of the Sun. This happens when the Sun is shining through a thin mist of hexagonal ice crystals falling with their principal axes vertical. If the principal axes are arranged randomly in a plane perpendicular to the Sun’s rays, the appearance is of a halo around the Sun.

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Timaeus of Locri was the central character in Plato’s
Timaeus
, but there was no real person by that name. Writings attributed to him cannot be considered examples of Pythagorean doctrine. They are an interpretation of Plato’s
Timaeus
, from the first century
B.C
. or the first century
A.D
.

*
Diogenes Laertius copied the excerpt not from the original but from an earlier author named Alexander Polyhistor who in turn–this was in the first half of the first century
B.C
.–copied it from a still older book.

^†
In view of all the other anachronisms in the
Notebooks
, scholars have ruled out the possibility that they were, after all, authentically early and primitively foreshadowed Aristotle’s cosmos.

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One clue has turned out to be a red herring: the suggestion that inclusion of superstition and “marvelous” events in a work represented more “primitive” thinking and dated the material earlier. Tales about a talking river or being in two places at the same time indicated that what you were reading was authentically early, so it was claimed. However, the late fourth century and the third, second, and first centuries
B.C
. and the early
A.D
. centuries were as accepting of magic, marvels, and portents as the fifth and sixth centuries
B.C
. had been—arguably more so. Such elements were expected in the biography of an important leader. Aristotle wrote during this period, when people may have been more ready to believe in a golden thigh than their fore-bears would have been at the time of Pythagoras. Clement of Alexandria, an eminent Christian scholar of the second and early third centuries
A.D
., described a “standard educational curriculum... astrology, mathematics, magic, and wizardry”—a quadrivium that would seem appropriate for Harry Potter’s Hogwarts School. “The whole of Greece,” Clement lamented, “prides itself on these as supreme sciences” (Clement of Alexandria, Stromateis 2.1.2. 3–4. Quoted in translation in J. Robert. Wright, ed.,
Ancient Christian Commentary on Scripture, Old Testament
IX [Downers Grove, Ill.: Intervarsity Press, p. 18]). For Diogenes Laertius, Porphyry, and Iamblichus, the fact that material included the miraculous did not invalidate the information or call the source into question. There was probably a mystical or magical element to the earliest Pythagoreanism, but late Greek, Alexandrian, and Roman writers were eager to report and exaggerate it. It is difficult to see through the veil of a superstitious age and judge how skeptical an earlier era was, but it is clear that one cannot decide that information was more authentically ancient simply because it included more of the “marvelous.”

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There was a legend about a Christian Philo, even a Bishop Philo, and a story in which he met the Apostle Peter.

^†
Shakespeare found the stories of
Antony and Cleopatra
,
Timon of Athens
, and
Coriolanus
in the
Lives
, and sometimes used Plutarch’s words virtually verbatim or changed them (as he read them in translation) only as much as was necessary to transform them into verse.

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Nicomachus was also intrigued by a pseudo-science called
gematria
that was not Pythagorean but originated with the ancient Babylonians and survived in ancient Greece and the Hellenistic period. In
gematria
, each letter of the alphabet had a numerical value. A word could be spelled in numbers. Sargon II, in the century before Pythagoras’ birth, had the wall of Khorsabad built to a measurement that was the numerical equivalent of his name—16,283 cubits. The name for the Gnostic divinity Abraxas had the numerical value of 365, the number of days in a solar year. Nicomachus did not claim that
gematria
was a Pythagorean practice, and it was not.

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Plotinus used and developed Numenius’ thoughts so extensively that he was accused of plagiarism. A colleague came to his rescue by writing an entire book to point out the differences between the two.

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This scale adds up to more than an octave, a problem easily corrected by changing the interval between Saturn and the stars to a half tone, as music theorists in later antiquity corrected Pliny. A half tone (half step) is the interval between one key and the next—black or white—on a piano.

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Along with all other pagan schools, the Academy would close in 526, two years after Boethius died, by order of the Byzantine emperor Justinian.

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Nestorian Christians were a group that originated in Asia Minor and Syria in the fifth century
A.D
. and stressed the human nature of Christ. There are still many thousands of them; today called the Church of the East, the Persian Church, or the Assyrian or Nestorian Church. Most Nestorians live in Iraq, Syria, and Iran.

^†
It is indicative of the cosmopolitan mix of religions and ideas in the Middle Ages in Islamic regions of the world that Hunayn’s writing, reflecting ancient pagan ideas and coming from a Christian who lived and worked in Islamic Baghdad, survived mainly because of a twelfth/thirteenth-century Hebrew translation by Judah al-Harizi.

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By “perfect number” they did not mean what the Pythagoreans had meant when they identified 10 as the perfect number. A perfect number by more modern standards (found already in Nicomachus) is a number the sum of whose divisors equals the number. The number 6 is the smallest perfect number: 1 + 2 + 3 = 6.

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A planet’s period is the time it takes to complete one orbit.

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Aurelian was reading from a mistranslation of the Book of Job.

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Regino’s description sounds very much like Aristotle’s, which means he must indeed have gotten it through Boethius. Regino lived before the reintroduction of Aristotle to Latin Europe.

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In the mid-twentieth century, there was still one expert, Vincenzo Capparelli, who was convinced that Pythagoras invented Arabic numerals (Vincenzo Capparelli,
La sapienza di Pitagora
[Padua: CEDAM, 1941]).

*
Most who used an abacus were still using Roman numerals, the English exchequer as late as the sixteenth century! (H. G. Koenigsberger,
Medieval Europe, 400–1500
[Harlow, England: Longman Group, 1987], p. 202.)

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T. S. Eliot echoed those sentiments when he suggested that to those who say we shouldn’t read the old authors since we know so much more than they did, we should answer, “And they are what we know.”

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The “Chaldean Oracles,” written in verse in the second century
A.D
. by a man named Julianus the Theurgist and his son, combined Babylonian and Persian beliefs with Platonic and neo-Platonic philosophy and became an important religious book for neo-Platonists.

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“Chaldean” in this case meaning Babylonian.

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The great Andrea Palladio was to write four.

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Alberti’s most important buildings included, in Florence, the Palazzo Rucellai, the Rucellai Chapel, the Annunziata, and the façade of the Maria Novella church; in Rimini, the Tempio Malatestiano; and in Mantua, the churches of San Sebastiano and San Andrea.

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Ecphantus the Pythagorean lived in the fourth century
B.C
. There is some suspicion that he may have been only a fictional character in one of Heracleides’ dialogues, but Copernicus thought he was a historical person, and most modern scholars tend to agree.

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A regular polygon is a flat shape in which all edges are the same length. For example: the triangle, square, pentagon, hexagon, etc. ad infinitum.

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A regular polyhedron is a solid shape in which all the edges have the same length and all the faces the same shape. The Pythagorean or Platonic solids are the regular polyhedra.

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When astronomers of Kepler’s time and earlier spoke of the “spheres,” they did not mean the planets. The Ptolemaic view of the cosmos had the planets traveling in transparent “crystalline spheres,” nested within one another like the layers of an onion and centered on the Earth. Though Kepler and Mästlin discussed spheres in their correspondence about Kepler’s new idea, Kepler (like his predecessor Tycho Brahe) did not believe there were actual glasslike spheres that one could crash through in a space vehicle. Thinking about them in a geometrical sense, not as physical reality, was nevertheless helpful in visualizing the movements of the planets.

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Depending on one’s definition of “planet,” Pluto and some other bodies that orbit the Sun may or may not have that status. Hence “eight or nine.”

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An example of a third on the piano is the interval from C to E (major third) or C to E-flat (minor third). An example of a sixth is the interval from C to A (major sixth) or C to A-flat (minor sixth). These are intervals that modern ears are most likely to hear as “beautiful” and easy to listen to.

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The Tychonic system had the Sun and the Moon orbiting the Earth, and all the other planets orbiting the Sun. It was the geometric equivalent of he Copernican system, but retained the unmoving Earth.

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Kepler’s first law of planetary motion: A planet moves in an elliptical orbit and the Sun is one focus of the ellipse. Kepler’s second law of planetary moton: A straight line drawn from a planet to the Sun sweeps out equal areas in equal times as the planet travels in its elliptical orbit.

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A half-step is the interval between any note on the piano and the one immediately to either side of it, regardless of whether that is a white or black key.

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Imagine you are standing across the corridor from a moving walkway in an airport. A man is walking along the walkway, from your left to your right, but he is going the wrong way and so is actually losing ground. Say he is walking at 5 miles per hour and the walkway is moving, in the opposite direction, at 10 miles per hour. From your vantage point, you see the combined movement, and the man appears to be moving 5 miles an hour toward the left. A woman is walking faster, 8 miles per hour, but also in the wrong direction. Eight miles per hour is not sufficient to avoid losing ground against the 10-mile-an-hour walkway that is moving in the opposite direction, so, again, from your vantage point, you see the combined motion, and this woman appears to be moving 2 miles per hour toward your left. You cannot be faulted for thinking that the man (who appears to be moving 5 miles per hour toward your left) is moving faster than the woman. If the walkway stopped you would find out what the true velocity of each one was, and your finding would contradict your initial impression. Likewise, Kepler concluded that if the daily rotation of the heavens had stopped, Pythagoras would have seen that Saturn is the slowest of the planets, and should be sounding the lowest tone.

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In German, dur in music still means “major”;
moll
is “minor.”