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

by Wasserman, James

  Applying both his hand and ear to the weights which he had hung—and by them confirming the proportion of the relations—he did ingeniously transfer the common result of the strings upon the cross-beam, to the bridge of an Instrument, which he called [“Stretched with strings”]. And as for stretching them proportionably to the weights, he did transfer that to an answerable screwing of the pegs. Making use of this foundation as an infallible rule, he extended the experiment to many kinds of instruments: cymbals, pipes, flutes, monochords, triangles, and the like. And he found, that this conclusion made by numbers was consonant without variation in all.

  That sound which proceeded from the number six, he named hypate; that which from the number eight, mese, being sesquitertia to the other; that from nine, paramese, being a tone sharper than the mese, viz. sesquioctave; that from twelve, neate. And supplying the middle spaces according to the diatonic kind, with proportional sounds, he so ordered the Octochord with convenient numbers: duple, sesquialtera, sesquitertia, and (the difference of these two last) sesquioctava.

  Arithmetical proportions of harmony

  From Thomas Stanley, The History of Philosophy

  Thus he found the progress by a natural necessity from the lowest to the highest according to the diatonical kind; from which again he did declare the Chromatic and Enharmonic kinds.

  CHAPTER 5

  THE DIVISION OF THE DIAPASON, ACCORDING TO THE DIATONIC KIND

  This Diatonic kind seems naturally to have these degrees and progresses: hemitone, tone and tone (half-note, whole note, and whole note). This is the system Diatessaron, consisting of two tones, and that which is called a hemitone. Then another tone being inserted, Diapente is made, being a system of three tones and a hemitone. Then, in order after this, there being another diatessaron, that is to say, another sesquitertia. So that in the more ancient Heptachord, all fourths from the lowest sound a diatessaron one to another: the hemitone taking the first, second, and third place, according to the progression in the Tetrachord. The Pythagorean Octochord is a conjunction of a system of the Tetrachord and the Pentachord, and that either jointly of two Tetrachords, or disjointly of two Tetrachords, separated from one another by a tone. Here, the procession will begin from the lowest. So that every fifth sound will make a diapente, the hemitone passing into four places, the first, the second, the third, and the fourth.

  CHAPTER 6

  THE CANON OF THE MONOCHORD

  Pythagoras, as Timaeus says, discovered the Canon of one Chord—that is the rule of the Monochord.607 Aristides relates that a little before he died, he exhorted his friends to play on the Monochord; thereby implying that the height which is in Music is to be received rather by the Intellect through numbers, than by the sense through the ears.608

  Duris (cited by Porphyry) mentions a brazen tablet set up in the temple of Juno by Arimnestus, son of Pythagoras, on which were graven, besides other arts, a Musical canon. This was afterwards taken away by Simon, a musician, who arrogated the canon to himself and published it as his own.609

  The division of the canon, says Theon, is made by the Tetractys in the Decad, which consists of a Monad, a Duad, a Triad, and a Tetrad—one, two, three, four.610 For it comprehends a sesquitertia, a sesquialtera, a duple, a triple, and a quadruple proportion. The section of the Pythagorean canon according to the intention of Pythagoras himself, not as Erastosthenes misunderstood it, or Thrasyllus (whose operation Theon sets down), but as Timaeus the Locrean (whom Plato also follows) to twenty-seven.611 Nicomachus mentions it, as intending to deliver it in his larger Treatise of Music.612 See also Euclid,613 Aristides Quintilianus,614 and others.

  CHAPTER 7

  INSTITUTION BY MUSIC

  Conceiving that the first institution of men was to be made by sense—so that a man might see those fair figures and forms, and might hear the most excellent music—he first began teaching music by songs and rhythms. By these, the cures of manners and passions were made, and the harmonies of the faculties of their souls were reduced to their primitive dispositions, and cures of distempers, both of body and mind, were invented by him.615

  And that music which was most worthy to be taken notice of, he reserved for his disciples—those which were called [“preparations”] and [“contact, touch”],† both by weight and by sound—he composed them harmonically in a strange way. He made commixtures of those tones which are called Diatonic, Chromatic, and Enharmonic. By means of these, he changed all the passions of the mind which were newly raised in them without reason—and which did procure griefs, angers, pities, unseemly loves, fears, all kind of desires, vexations, appetites, softnesses, idlenesses, and impetuosities—correcting and directing every one of these towards virtue, by convenient harmonies, as by certain effectual medicines.

  And at night when his disciples went to sleep, he delivered them from all the noises and troubles of the day, and purified the perturbations of their minds, and rendered their sleeps quiet with good dreams and predictions. And when they rose again from their beds, he freed them from the drowsiness of the night, from faintness and sluggishness, by certain proper songs, either set to the Lute or some high voice.

  As for himself, he never played on an instrument or anything. But he had it within him; and by an inconceivable kind of divinity, he applied his ears and mind unto the harmony of the world, which he alone did understand. And he understood the universal harmony and consent of the spheres, and those stars that move in them, which make a more full and excellent music than mortals. This they do by reason of their motion, and of unequal and differing swiftnesses and bignesses, overtaking one another, all of which are ordered and disposed in a most musical proportion one towards another. Being beautified with various perfections wherewith he was endowed, as having likewise the command of an orderly mind, as we may say, exercising it—he framed some representations of these sounds to exhibit them as much as was possible, imitating (that music) chiefly by instruments or the voice alone.

  For he conceived that to himself only, of all upon the earth, were intelligible and audible the universal sounds from the natural fountain and root. And he thought himself worthy to be taught and to learn, and to be assimilated by desire and imitation to the celestials, as one that was organized in the parts of the body by the deity which begot him. But it was sufficient for other men that they, always looking upon him, and such things as they received from him, be benefited by images and examples; since they were not able to lay hold on the first clear archetypes of all things. As to them who cannot look upon the Sun by reason of its splendor, we show the eclipse either in a pond of water, or by some bored pitched thing, or by some dark-colored glass, searing the weakness of their eyes, and framing another way of perception instead of looking on it, to those who love such things, though something inferior.

  This, Empedocles seems to imply, concerning the extraordinary and divine constitution of Pythagoras above others, when he said:

  Amongst these was one in things sublimest, skilled

  His mind with all the wealth of learning filled.

  He sought whatever Sages did invent;

  And while his thoughts were on this work intent,

  All things that are, he easily surveyed,

  And search through ten or twenty ages made.

  Intimating by sublimest things, he surveyed all things that are. And the wealth of the mind and the like, the exquisite and accurate constitution of Pythagoras was beyond others, both for body and mind, in seeing, hearing, and understanding.

  CHAPTER 8

  MEDICINE BY MUSIC

  Pythagoras conceived that music contributed much to health if used properly; for he was accustomed to make use of this purification, not perfunctorily.616 This he called medicine by music, which kind of melody he exercised about the springtime. He seated him who played on the lute in the midst, and those who could sing sat round about him; and so he playing, they made a consort of some excellent pleasant verses, wherewith they seemed exhilarated and decently composed.
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  They likewise at another time made use of music as of a medicine; and there were certain pleasant verses framed tending much against the affections and diseases of the mind, and against the dejections and corrodings of the same. Moreover, he composed other music against anger and malice, and all such disorders of the mind. There was also another kind of music and song invented, against unlawful desires. He likewise used dancing. He used no musical instrument but the lute. Wind-instruments he conceived to have an ignoble sound, and to be only fit for the common people, but nothing generous.

  He likewise made use of the words of Homer and Hesiod for the rectifications of the mind. It is reported that Pythagoras, by a spondaic verse out of the works617 (perhaps of Hesiod, whose poem bears that title, [“work”]†), by a player on the flute, assuaged the madness of a young man of Tauromenium. He, being drunk, and having employed all the night lasciviously with his mistress, was going about to burn the door of his rival's house; for he was exasperated and enflamed by the Phrygian mood. But Pythagoras, who was at that time busied in observing the stars, immediately appeased and reclaimed him by persuading the piper to change his air into the spondaic mood. Whereupon the young man being suddenly composed went quietly home—who but a little before would by no means hear the least exhortation from Pythagoras, but threatened and reviled him. In like manner Empedocles, when a young man, drew his sword upon Anchitus his host (for that he had in public judgment condemned his father to death). Empedocles was about to have killed him, when Pythagoras straightway changing his tune, sung out of Homer:

  Nepenthe calming anger, easing grief.

  And by that means freed Anchitus his host from death, and the young man from the crime of murder; who from thence forward became one of his disciples, eminent amongst them.

  Moreover the whole school of Pythagoras made that which is called [“preparedness”], and [“musical combination”], and [“contact”],† by certain verses suitable thereto and proper against the contrary affections, profitably diverting the constitutions and dispositions of the mind. For when they went to bed and resigned themselves to rest, they purified their minds from the troubles and busy noises of the day by some songs and proper verses, whereby they rendered their sleeps pleasant and quiet, and little troubled with dreams. And those dreams which they had were good. In the morning, when they arose from the common relief of sleep, they expelled drowsiness and sleepiness of the head with other songs.

  Sometimes also without pronouncing verses, they expelled some affections and diseases and brought the sick to health, by charming them.† And from hence it is probable that the word “Epode” [an incantation] came to be used. After this manner, Pythagoras instituted a most profitable correction of manners and life by music. Hitherto Iamblichus.

  All of which is ratified by other testimonies. That they had verses against the affections of the mind—grief, anger, lust—is related also by Seneca, who says that Pythagoras composed the troubles of his soul by the lute.618 And Cicero, that the Pythagoreans used to deliver verses and some precepts, and to reduce the mind from intensity of thoughts to tranquility by songs and instruments.619 To which effect Aelian relates to Clinias the Pythagorean, that if at any time he perceived himself inclining to anger, before it took full possession of him, he played upon the lute; and to those who asked him why he did so, he answered, “Because I am calmed.”620

  That Pythagoras danced, Porphyry confirms, saying he danced some dances, which he conceived to confer agility and health to the body.621

  That he disallowed flutes and wind-instruments appears in Aristides Quintilianus, who says he advised his disciples to refrain from permitting their ears to be defiled with the sound of the flute; but on the contrary, to purify the irrational impulsions of the soul by solemn songs on the lute.622

  That he made use of Homer and Hesiod for rectification of the mind is thus related by Porphyry.623 He had morning exercises at his own house, composing his soul to the lute, and singing some old paeans of Thales.624 He likewise sung some verses of Homer and Hesiod whereby the mind seemed to be rendered more sedate.

  The story of the young man is confirmed by Ammonius.625 By Cicero, this is related: Whenas some young men being drunk, and irritated by the music of flutes, would have broken open the door of a modest matron's house, Pythagoras had the woman-piper play a spondaic tune; which as soon as she did, their raging petulancy was allayed by the slowness of the mood, and solemness of the tune.626 St. Basil relates another story to the same purpose: that Pythagoras meeting with some that came from a feast drunk, bid the piper (the musician at that feast) to change his tune, and to play a Doric air; wherewith they were so brought to themselves that they threw away their garlands, and went home ashamed.627

  That evening and morning his disciples used music to compose their minds is affirmed by many others. Quintilian wrote, “It was the custom of the Pythagoreans as soon as they waked, to excite their souls with the lute, that they might be the readier for action; and before they went to sleep, to soften their minds by it.”628 Plutarch says, “The music of the lute the Pythagoreans used before they went to sleep, thereby charming and composing the passionate and irrational part of the soul.”629 Censorinus wrote that Pythagoras, so that his mind might be continually seasoned with divinity, used (as they say) to sing before he went to sleep and as soon as he waked.630

  As for the several moods which in musical compositions were observed by the Ancients for moving particular passions, there is a remarkable fragment of Damon the musician cited by Aristides.631

  GEOMETRY

  Pythagoras (says Iamblichus) is reported to have been much addicted to Geometry, for amongst the Egyptians, of whom he learned it, there are many geometrical problems.632 The most learned Egyptians were continually, for many ages of gods and men, required to measure their whole country by reason of the overflowing and decrease of the Nile—whence it is called Geometry.633

  Some there are who ascribe all theorems concerning Lines jointly to the Egyptians and the Chaldeans; and all these, they say, Pythagoras took, and augmenting the science, explained them accurately to his disciples. Proclus affirms that Pythagoras first advanced the geometrical part of learning into a liberal science, considering the principles more sublimely (than Thales, Ameristus, and Hippias, his predecessors in this study) and thoroughly investigated the theorems immaterially and intellectually.634 Timaeus says that he first perfected geometry, the elements whereof (as Anticlides affirms), were invented by Moeris.635 Aristoxenus says that Pythagoras first introduced measures and weights amongst the Grecians.636

  CHAPTER 1

  OF A POINT, LINE, SUPERFICIES AND SOLID

  Pythagoras asserted a Point to be correspondent in proportion to a unit; a Line to two; a Superficies to three: a Solid to four. 637

  The Pythagoreans define a Point as a Monad having Position.638

  A Line, being the second, and constituted by the First Motion from indivisible nature, they called Duad.639

  A Superficies they compared to the number three, for that is the first of all causes which are found in figures: for a circle, which is the principle of all round figures, occultly comprises a triad in center, space, and circumference.640 But a triangle, which is the first of all rectilinear figures, is manifestly included in a ternary, and receives its form according to that number.

  Hence the Pythagoreans affirm, that the triangle is simply the Principle of Generation, and of the formation of things generable. Whereupon Timaeus says that all proportions, natural as well as of the constitution of elements, are triangular; because they are distant by a threefold interval and are collective of things every way divisible. Triangles are variously permutable and are replenished with material infinity, and represent the natural conjunctions of bodies dissolved. As triangles are comprehended by three right lines, they also have angles which collect the multitude of lines, and give the additional property of an angle and conjunction to them.641

  With reason therefore did Philolaus de
dicate the angle of a triangle to four gods: Saturn, Pluto, Mars, and Bacchus—comprehending in these the whole quadripartite ornament of elements coming down from heaven or from the four quarters of the Zodiac. For Saturn constitutes an essence wholly humid and frigid; Mars wholly fiery; Pluto comprises all terrestrial life; Bacchus predominates over humid and hot generation, of which wine is a sign being humid and hot. All these differ in their operations upon second bodies, but are united to one another, for which reason Philolaus collected their union according to one angle.