Pythagoras: His Life and Teaching, a Compendium of Classical Sources

by Wasserman, James

  Panarmonia,543 because of its excellent convenience.544

  Cadmaea, Mother, Rhea, [“making weak”],† Cibele, Dindymens, [“guardian of the city”],† Love, Friendship, Council, Prudence, Orcia, Themis, Law, [“untimely born”],† Euterpe the Muse, [“stability”],†” [“placing in position”],† 545 Neptune.546

  Justice, because it is first resolved into numbers, especially equal.547

  CHAPTER 13

  THE ENNEAD

  The Ennead is the first square of an odd number. Its names are these:

  Ocean, Horizon, because number has nothing beyond it, but it revolves all within it.548

  Prometheus, because it suffers no number to exceed it, and justly being a perfect ternary.549

  Concord,550 Perasia,551 Halius,552 because it does not permit the consent of number to be dispersed beyond it, but collects it.553

  [“reconciliation”]† because of the revolution to Monad.554

  [“similitude”], because it is the first odd triangle.555

  Vulcan, because to it, as fellow ruler and relation, there is no return.556

  Juno, because the sphere of the Air has the ninth place.557

  Sister and Wife to Jupiter, from conjunction with unity.558

  [“far-darting”],† because there is no shooting beyond it.559

  Paean, Nysseis, Agyica, Ennalios, Agelia, Tritogegenia, Suada, Curetis, Proserpina, Hyperion, Terpsichore the Muse.560

  [“bringing to the end”],† [“perfection”], because nine months complete the infant.

  CHAPTER 14

  THE DECAD

  Ten, according to the Pythagoreans, is the greatest number—as well for that it is the Tetractys, as that it comprehends all arithmetical and harmonical proportions.561 Pythagoras said that ten is the nature of number: because all nations, Greeks and Barbarians, reckon to it; and when they arrive at it, return to the Monad.562

  Names of the Decad:

  World, because according to the Decad all things are ordered in general and particular.563 The Decad comprehends all numbers, the world all form;564 for the same reason it is termed also Sphere.565

  Heaven,566 because it is the most perfect term of number, as heaven is the receptacle of all things.567 The Decad being a perfect number, the Pythagoreans desired to apply to it those things which are contained in Heaven—where finding but nine (the orbs, the seven planets, and the heaven of Fixed Stars, with the earth), they added an Antichthon (another earth opposite to this) and made Ten; by this means they accommodated them to the Decad.568

  Fate, 569 because there is no property neither in numbers nor beings, according to the composition of number, which is not seminally contained in the Decad.570

  Age.571

  Power,572 from the command it has over all other numbers.573

  Faith, Necessity.574

  Atlas, for as Atlas is fabled to sustain heaven with his shoulders, so the Decad holds all the spheres as the diameter of them all.575

  Unwearied, God, Phanes, Sun, Urania, Memory, Mnemosyne.576

  First Square, because it is made of the first four numbers: one plus two plus three plus four.577

  [“key-bearer”], as the magazine and confinement of all proportions,578 or [“branch-bearer”], because other numbers branch out of it.579

  [“the absolute”], because it perfects all number, comprehends within itself all the nature of even and odd, moved and unmoved, good and ill.580

  CHAPTER 15

  DIVINATION BY NUMBERS

  Upon the near affinity which Pythagoras (following Orpheus) conceived to be between the gods and numbers, he collected a kind of Arithmancy. This he not only practiced himself, but communicated to his disciples—as is manifest from Iamblichus, who cites this fragment of the Sacred Discourse, a book ascribed to him.

  “Concerning the gods of Pythagoras, son of Mnesarchus, I learned this when I was initiated at Libeth in Thrace, Aglaophemus administering the rites to me. Orpheus, son of Calliope, instructed by his mother in the Pangaean mountain, said that number is an eternal substance, the most provident principle of the universe: heaven, and earth, and middle nature; likewise the root of divine beings, and of gods and daemons.” 581

  Hence (says Iamblichus) it is manifest that Pythagoras received from the traditions of Orpheus the doctrine that numbers hold the determinate essence of the gods. By these numbers he framed a wonderful system of divination and service of the gods. This had the closest affinity to numbers, as may be evinced from hence (for it is requisite to give an instance for confirmation of what we say).

  The student of Pythagoras, Abaris, performed those kinds of sacrifices to which he was accustomed, and diligently practiced divination after the ways of the Barbarians by victims (principally of cocks, whose entrals they conceived to be most exact for inspection). Pythagoras, not willing to take him away from his study of truth; yet, in order to direct him by a safer way, without blood and slaughter (moreover esteeming the cock sacred to the Sun), taught Abaris to find out all truth by the science of arithmetic.582 Thus says Iamblichus, who writes elsewhere that Pythagoras, instead of the art of divining by sacrifices, taught that kind of prediction which is by numbers, conceiving that to be more sacred and divine, and more agreeable to the celestial nature of the gods.

  This hint some have taken to impose upon the world, under the name of Pythagoras, an Onomantic kind of arithmetic—assigning particular numbers to the letters of the alphabet, to the planets, to the days of the week, and to the signs of the Zodiac. They thereby resolve questions concerning nativities, victory, life or death, journeys, prosperity or adversity. Such a system is set down by Fludd,583 who adds that Apollonius has delivered another way of divination according to the Pythagorean doctrine; affirming that future things may be prognosticated by virtue of a wheel invented by Pythagoras. Hereby is treated of life and death, of fugitives, of litigious business, of victories, of the sex of children unborn, and infinite others of the like kind. But concerning the exposition of the wheel, and the true position of numbers, therein the ancient authors have written very inconstantly. So that the truth of its composition cannot be comprehended otherwise than by conjecture. What ancient authors he means I know not. The citation of Apollonius I doubt to be no less an assumption than the wheel itself, which Trithemius and others acknowledge to be an invention of later times.584

  MUSIC

  The Pythagoreans define music as an apt composition of contraries, a union of many, and consent of differents.585 For it not only coordinates rhythms and modulation, but all manner of systems. Its end is to unite, and aptly conjoin. God is the reconciler of things discordant, and his chief work—according to music and medicine—is to reconcile enmities. In music consists the agreement of all things, and the aristocracy of the universe. For what is harmony in the world, in a city is good government, in a family temperance.

  Of many sects (says Ptolemy) that were conversant about harmony, the most eminent were two: the Pythagorean and Aristoxenian.586 Pythagoras judged it by reason, Aristoxenus by sense.587 The Pythagoreans, not crediting the relation of hearing in all those things wherein it is requisite, adapted reasons to the differences of sounds, contrary to those which are perceived by the senses. So that by this criterion (reason), they gave occasion of calumny to such as were of a different opinion.588

  Hence the Pythagoreans named that which we now call Harmonic, “Canonic”; not from the canon or instrument, as some imagine, but from rectitude—since reason finds out that which is right by using harmonic canons or rules.589 Even of all sorts of instruments framed by harmonic rules (pipes, flutes, and the like), they call the exercise canonic; which, although it be not canonic, yet is so termed because it is made according to the reasons and theorems of canonic. The instrument therefore seems to be so denominated from its canonic affection.

  A follower of canonic doctrine is a “harmonic” who is conversant by ratiocination about that which consists of harmony. Musicians and harmonics differ. Musicians are th
ose harmonics who begin from sense; but canonics are Pythagoreans, who are also called harmonics. Both sorts are termed by the general name, Musicians.

  CHAPTER 1

  VOICE, ITS KINDS

  Of human voice, those of the Pythagorean school said that there are (as of one genus) two species. One they properly named Continuous, the other Diastematic (intermissive), framing appellations from the accidents pertaining to each. The diastematic they conceived to be that which is sung and rests upon every note, and manifests the mutation which is in all its parts. It is free from confusion and divided and disjoined by the magnitudes which are in the several sounds, as accumulated but not mixed up. The parts of the voice, being applied mutually to one another, may easily be separated and distinguished, and are not destroyed together. Such is the musical kind of voice, which to the knowing, manifests all sounds of what magnitude everyone participates. For if a man use it not after this manner, he is not said to sing, but to speak. 590

  The other kind they conceived to be continuous, by which we discourse one to another, and read. We are not constrained to use any manifest distinct tensions of sounds, but to connect the discourse till we have finished that which we intended to speak. For if any man, in disputing or apologizing or reading, makes distinct magnitudes in the several sounds, taking off and transferring the voice from one to another, he is not said to read but to sing.

  Human voice, having in this manner two parts, they conceived that there are two places which each in passing possesses. The place of continuous voice—which is by nature infinite in magnitude—receives its proper term from that wherewith the speaker began until he ends; that is the place from the beginning of his speech to his conclusive silence; so that the variety thereof is in our power. But the place of diastematic voice is not in our power, but natural. And this likewise is bounded by different effects. The beginning is that which is first heard, the end that which is last pronounced. For from thence we begin to perceive the magnitude of sounds, and their mutual commutations, from whence first our hearing seems to operate.

  Whereas it is possible there may be some more obscure sounds perfected in nature which we cannot perceive or hear. As for instance, in things weighed there are some bodies which seem to have no weight, such as straws, bran, and the like. But when, as by the adding together of such bodies some beginning of ponderosity appears, then we say they first come within the compass of static. So, when a low sound increases by degrees, that which first of all may be perceived by the ear we make the beginning of the place which musical voice requires.

  CHAPTER 2

  FIRST MUSIC IN THE PLANETS

  The names of sounds, in all probability, were derived from the seven stars, which move circularly in the heavens and compass the earth.591 (The circular motion of these bodies must of necessity cause a sound, for air being struck from the intervention of the blow sends forth a noise. Nature herself constraining that the violent collision of two bodies should end in sound.592)

  Now (say the Pythagoreans) all bodies which are carried round with noise—one yielding and gently receding to the other—must necessarily cause sounds different from each other, in the magnitude and swiftness of voice and in place. These, according to the reason of their proper sounds, or their swiftness, or the orbs of repressions, in which the impetuous transportation of each is performed—are either more fluctuating, or on the contrary, more reluctant. But these three differences of magnitude, celerity, and local distance, are manifestly existent in the planets. These planets are constantly with sound circling around through the aetherial diffusion, whence every one is called [star], as void of station; and , always in course; whence God and Aether are called and .593

  Moreover the sound which is made by striking the air induces into the ear something sweet and musical, or harsh and discordant. For if a certain observation of numbers moderates the blow, it effects a harmony consonant to itself; but if it be haphazard and not governed by measures, there proceeds a troubled unpleasant noise which offends the ear.594 Now in heaven nothing is produced casually or randomly; but all things there proceed according to divine rules and settled proportions. Whence it may be irrefutably inferred that the sounds which proceed from the conversion of the Celestial Spheres are musical. For sound necessarily proceeds from motion—and the proportion, which is in all divine things causes the harmony of this sound. This Pythagoras, first of all the Greeks, conceived in his mind. He understood that the Spheres sounded something concordant because of the necessity of proportion which never forsakes celestial beings.

  From the motion of Saturn, which is the highest and furthest from us, the gravest sound in the diapason concord is called Hypate, because signifys highest. From the Lunary, which is the lowest and nearest the earth, neate, for signifys lowest.595 From those which are next these, viz. from the motion of Jupiter who is under Saturn, parypate; and of Venus, who is above the Moon, paraneate. Again, from the middle, which is the Sun's motion, the fourth from each part, mese, which is distant by a diatessaron in the Heptachord from both extremes according to the ancient way; as the Sun is the fourth from each extreme of the seven planets, being in the middle. Again, from those which are nearest the Sun on each side: from Mars who is placed between Jupiter and the Sun, hypermese, which is likewise termed lichanus; and from Mercury who is placed between Venus and the Sun, paramese.

  Pythagoras, by Musical proportion, calls that a Tone by how much the Moon is distant from the Earth; from the Moon to Mercury the half of that space; and from Mercury to Venus almost as much. From Venus to the Sun sesqidulple; from the Sun to Mars a tone (that is as far as the Moon is from the Earth); from Mars to Jupiter half; and from Jupiter to Saturn half; and thence to the Zodiac sesquiduple. Thus there are made seven tones: which they call a Diapason harmony, that is an universal concord, in which Saturn moves in the Doric mood, Jupiter in the Phrygian, and in the rest the like.596

  The sounds made by the seven planets and the Sphere of Fixed Stars, and that which is above us (termed by them Antichthon), Pythagoras affirmed to be the Nine Muses. But the composition and symphony, and, as it were, connection of them all—whereof as being eternal and unbegotten, each is a part and portion—he named Mnemosyne.

  CHAPTER 3

  THE OCTOCHORD

  Now Pythagoras, first of all, left the middle sound by conjunction, being itself compared to the two extremes, should render only a diatessaron harmony, both to the neate and to the hypate. But that we might have greater variety, the two extremes making the fullest concord each to other, that is to say the concord of diapason, which consists in a double proportion.597 Inasmuch as it could not be done by two Tetrachords, he added an eighth sound, inserting it between the mese and paramese—setting it from the mese a whole tone, and from the paramese a semitone. So that which was formerly the paramese in the Heptachord is still the third from the neate, both in name and place. But that which was now inserted is the fourth from the neate, and has a harmony unto it of diatessaron—which before, the mese had unto the hypate.598

  But the tone between them, that is the mese, and the inserted called the paramese, instead of the former, to whichever Tetrachord it be added, whether to that which is at the hypate, being of the lower; or to that of the neate, being of the higher; will render diapente concord. This is either way a system consisting both of the Tetrachord itself, and the additional tone; as the diapente-proportion (viz. sesquialtera) is found to be a system of sesquitertia, and sesquioctava; the Tone therefore is sesquioctava.599 Thus the interval of four chords, and of five, and of both conjoined together, called diapason, and the tone inserted between the two Tetrachords, being after this manner apprehended by Pythagoras, were determined to have this proportion in numbers.

  CHAPTER 4

  THE ARITHMETICAL PROPORTIONS OF HARMONY

  Pythagoras is said to have first found out the proportion and concord of sounds one to another: the Diatessaron in sesquitertia, the Diapente in sesquialtera, the Diapason in duple.600 The occasi
on and manner is related by Censorinus,601 Boethius,602 Macrobius,603 and others; but more exactly by Nicomachus604 thus:

  Being in an intense thought, whether he might invent any instrumental help for the ear, solid and infallible—such as the sight has by a compass, and a rule, and by a diopter; or the touch by a balance, or by the invention of measures—as he passed by a smith's shop, by a happy chance he heard the iron hammers striking upon the anvil, and rendering sounds most consonant one to another in all combinations except one. He observed in them these three concords: the diapason, the diapente, and the diatessaron. But that which was between the diatessaron and the diapente, he found to be a discord in itself, though otherwise useful for the making up of the greater of them (the diapente).

  Apprehending this to come to him from God as a most happy thing, he hastened into the shop. By various trials he found the difference of the sounds to be according to the weight of the hammers—and not according to the force of those who struck, nor according to the fashion of the hammers, nor according to the turning of the iron which was in beating out. Having taken exactly the weight of the hammers,605 he went straightaway home. He tied four strings of the same substance, length, swiftness, and twist 606 to a beam on one side of the room, and then extended and fastened the other end of the strings to the wall on the other side of the room (lest any difference might arise from thence, or might be suspected to arise from the properties of several beams). Upon each of them he hung a different weight, fastening it at the lower end, and making the length of the strings altogether equal. Then striking the strings by two at a time interchangeably, he found out the aforesaid concords, each in its own combination.

  For that which was stretched by the greatest weight, in respect of that which was stretched by the least weight, he found to sound a diapason. The greatest weight was of twelve pounds, the least of six. Thence he determined that the diapason did consist in double proportion, which the weights themselves did show. Next he found that the greatest to the least but one, which was of eight pounds, sounded a diapente. Whence he inferred this to consist in the proportion called sesquialtera, in which proportion the weights were one to another. But unto that which was less than itself in weight, yet greater than the rest, being of nine pounds, he found it to sound a diatessaron. He discovered that proportionably to the weights, this concord was sesquitertia, which string to nine pounds is naturally sesquialtera to the least. For nine to six is so (viz. sesquialtera) as the least but one, which is eight, was to that which had the weight six, in proportion sesquitertia. And twelve to eight is sesquialtera. And that which is in the middle between diapente and diatessaron, whereby diapente exceeds diatessaron, is confirmed to be in sesquioctava proportion, in which nine is to eight. The system of both was called Diapente, that is, both of the diapente and diatessaron joined together, as duple proportion is compounded of sesquialtera and sesquitertia, such as are two, eight, six. Or on the contrary, of diatessaron and diapente, as duple proportion is compounded of sesquitertia and sesquialtera, as twelve, nine, six being taken in that order.