(399) under the circumstances, and acquiescing in the event. These two harmonies I ask you to leave; the strain of necessity and the strain of freedom, the strain of the unfortunate and the strain of the fortunate, the strain of courage, and the strain of temperance; these, I say, leave. And these, he replied, are the Dorian and Phrygian harmonies of which I was just now speaking. Then, I said, if these and these only are to be used in our songs and melodies, we shall not want multiplicity of notes or a panharmonic scale? I suppose not. Then we shall not maintain the artificers of lyres with three corners and complex scales, or the makers of any other many-stringed curiously-harmonised instruments? Certainly not. But what do you say to flute-makers and flute-players? Would you admit them into our State when you reflect that in this composite use of harmony the flute is worse than all the stringed instruments put together; even the panharmonic music is only an imitation of the flute? Clearly not. There remain then only the lyre and the harp for use in the city, and the shepherds may have a pipe in the country. That is surely the conclusion to be drawn from the argument. The preferring of Apollo and his instruments to Marsyas and his instruments is not at all strange, I said. Not at all, he replied. And so, by the dog of Egypt, we have been unconsciously purging the State, which not long ago we termed luxurious. And we have done wisely, he replied. Then let us now finish the purgation, I said. Next in order to harmonies, rhythms will naturally follow, and they should be subject to the same rules, for we ought not to seek out complex systems of metre, or metres of every kind, but rather to discover what rhythms are the expressions of

(2) Employing this method, therefore, as a basis, and as it were an infallible rule, he afterwards extended the experiment to various instruments; viz. to the pulsation of patellæ or pans, to pipes and reeds, to monochords, triangles, and the like. And in all these he found an immutable concord with the ratio of numbers. But he denominated the sound which participates of the number 6 hypate : that which participates of the number 8 and is sesquitertian, mese ; that which participates of the number 9, but is more acute by a tone than mese, he called paramese , and epogdous ; but that which participates of the dodecad, nete . Having also filled up the middle spaces with analogous sounds according to the diatonic genus, he formed an octochord from symphonious numbers, viz. from the double, the sesquialter, the sesquitertian, and from the difference of these, the epogdous.

And thus he discovered the [harmonic] progression, which tends by a certain physical necessity from the most grave [i. e. flat] to the most acute sound, according to this diatonic genus. For from the diatonic, he rendered the chromatic and enharmonic genus perspicuous, as we shall some time or other show when we treat of music. This diatonic genus, however, appears to have such physical gradations and progressions as the following; viz. a semitone, a tone, and then a tone; and this is the diatessaron, being a system consisting of two tones, and of what is called a semitone. Afterwards, another tone being assumed, viz. the one which is intermediate, the diapente is produced, which is a system consisting of three tones and a semitone.

In the next place to this is the system of a semitone, a tone, and a tone, forming another diatessaron, i. e. another sesquitertian ratio. So that in the more ancient heptachord indeed, all the sounds, from the most grave, which are with respect to each other fourths, produce every where with each other the symphony diatessaron; the semitone receiving by transition, the first, middle, and third place, according to the tetrachord. In the Pythagoric octachord, however, which by conjunction is a system of the tetrachord and pentachord, but if disjoined is a system of two tetrachords separated from each other, the progression is from the most grave sound. Hence all the sounds that are by their distance from each other fifths, produce with each other the symphony diapente; the semitone successively proceeding into four places, viz. the first, second, third, and fourth. After this manner, therefore, it is said that music was discovered by Pythagoras. And having reduced it to a system, he delivered it to his disciples as subservient to every thing that is most beautiful.

(1) Since, however, we are narrating the wisdom employed by Pythagoras in instructing his disciples, it will not be unappropriate to relate that which is proximate in a following order to this, viz. how he invented the harmonic science, and harmonic ratios. But for this purpose we must begin a little higher. Intently considering once, and reasoning with himself, whether it would be possible to devise a certain instrumental assistance to the hearing, which should be firm and unerring, such as the sight obtains through the compass and the rule, or, by Jupiter, through a dioptric instrument; or such as the touch obtains through the balance, or the contrivance of measures;—thus considering, as he was walking near a brazier’s shop, he heard from a certain divine casualty the hammers beating out a piece of iron on an anvil, and producing sounds that accorded with each other, one combination only excepted.

But he recognized in those sounds, the diapason, the diapente, and the diatessaron, harmony. He saw, however, that the sound which was between the diatessaron and the diapente was itself by itself dissonant, yet, nevertheless, gave completion to that which was the greater sound among them. Being delighted, therefore, to find that the thing which he was anxious to discover had succeeded to his wishes by divine assistance, he went into the brazier’s shop, and found by various experiments, that the difference of sound arose from the magnitude of the hammers, but not from the force of the strokes, nor from the figure of the hammers, nor from the transposition of the iron which was beaten. When, therefore, he had accurately examined the weights and the equal counterpoise of the hammers, he returned home, and fixed one stake diagonally to the walls, lest if there were many, a certain difference should arise from this circumstance, or in short, lest the peculiar nature of each of the stakes should cause a suspicion of mutation.

Afterwards, from this stake he suspended four chords consisting of the same materials, and of the same magnitude and thickness, and likewise equally twisted. To the extremity of each chord also he tied a weight. And when he had so contrived, that the chords were perfectly equal to each other in length, he afterwards alternately struck two chords at once, and found the before-mentioned symphonies, viz. a different symphony in a different combination. For he discovered that the chord which was stretched by the greatest weight, produced, when compared with that which was stretched by the smallest, the symphony diapason. But the former of these weights was twelve pounds, and the latter six. And, therefore, being in a duple ratio, it exhibited the consonance diapason; which the weights themselves rendered apparent.

But again, he found that the chord from which the greatest weight was suspended compared with that from which the weight next to the smallest depended, and which weight was eight pounds, produced the symphony diapente. Hence he discovered that this symphony is in a sesquialter ratio, in which ratio also the weights were to each other. And he found that the chord which was stretched by the greatest weight, produced, when compared with that which was next to it in weight, and was nine pounds, the symphony diatessaron, analogously to the weights. This ratio, therefore, he discovered to be sesquitertian; but that of the chord from which a weight of nine pounds was suspended, to the chord which had the smallest weight [or six pounds,] to be sesquialter.

For 9 is to 6 in a sesquialter ratio. In like manner, the chord next to that from which the smallest weight depended, was to that which had the smallest weight, in a sesquitertian ratio, [for it was the ratio of 8 to 6,] but to the chord which had the greatest weight, in a sesquialter ratio [for such is the ratio of 12 to 8.] Hence, that which is between the diapente and the diatessaron, and by which the diapente exceeds the diatessaron, is proved to be in an epogdoan ratio, or that of 9 to 8. But either way it may be proved that the diapason is a system consisting of the diapente in conjunction with the diatessaron, just as the duple ratio consists of the sesquialter and sesquitertian, as for instance, 12, 8, and 6; or conversely, of the diatessaron and the diapente, as in the duple ratio of the sesquitertian and sesquialter ratios, as for instance 12, 9, and 6.

After this manner, therefore, and in this order, having conformed both his hand and his hearing to the suspended weights, and having established according to them the ratio of the habitudes, he transferred by an easy artifice the common suspension of the chords from the diagonal stake to the limen of the instrument, which he called chordotonon . But he produced by the aid of pegs a tension of the chords analogous to that effected by the weights.

(27) Many early instruments had seven Strings, and it is generally conceded that Pythagoras was the one who added the eighth string to the lyre of Terpander. The seven strings were always related both to their correspondences in the human body and to the planets. The names of God were also conceived to be formed from combinations of the seven planetary harmonies. The Egyptians confined their sacred songs to the seven primary sounds, forbidding any others to be uttered in their temples. One of their hymns contained the following invocation: "The seven sounding tones praise Thee, the Great God, the ceaseless working Father of the whole universe." In another the Deity describes Himself thus: "I am the great indestructible lyre of the whole world, attuning the songs of the heavens. (See Nauman's History of Music.)

(14) Pythagoras evinced such a marked preference for stringed instruments that he even went so far as to warn his disciples against allowing their ears to be defiled by the sounds of flutes or cymbals. He further declared that the soul could be purified from its irrational influences by solemn songs sung to the accompaniment of the lyre. In his investigation of the therapeutic value of harmonics, Pythagoras discovered that the seven modes--or keys--of the Greek system of music had the power to incite or allay the various emotions. It is related that while observing the stars one night he encountered a young man befuddled with strong drink and mad with jealousy who was piling faggots about his mistress' door with the intention of burning the house. The frenzy of the youth was accentuated by a flutist a short distance away who was playing a tune in the stirring Phrygian mode. Pythagoras induced the musician to change his air to the slow, and rhythmic Spondaic mode, whereupon the intoxicated youth immediately became composed and, gathering up his bundles of wood, returned quietly to his own home.

(2) We must rather, therefore, say, that sounds and melodies are appropriately consecrated to the Gods. There is, also, an alliance in these sounds and melodies to the proper orders and powers of the several Gods, to the motions in the universe itself, and to the harmonious sounds which proceed from the motions. Conformably, therefore, to such like adaptations of melodies to the Gods, the Gods themselves become present. For there is not any thing which intercepts; so that whatever has but a casual similitude to, directly participates of, them . A perfect possession, likewise, immediately takes place, and a plenitude of a more excellent essence and power. Not that the body and the soul are in each other, and sympathize, and are copassive with the melodies; but because the inspiration of the Gods is not separated from divine harmony, but is originally adapted and allied to it, on this account it is participated by it in appropriate measures. Hence also, it is excited and restrained according to the several orders of the Gods. But this inspiration must by no means be called an ablation, purgation, or medicine. For it is not primarily implanted in us from a certain disease, or excess, or redundance; but the whole principle and participation of it are supernally derived from the Gods.

(398) And which are the harmonies expressive of sorrow? You are musical, and can tell me. The harmonies which you mean are the mixed or tenor Lydian, and the full-toned or bass Lydian, and such like. These then, I said, must be banished; even to women who have a character to maintain they are of no use, and much less to men. Certainly. In the next place, drunkenness and softness and indolence are utterly unbecoming the character of our guardians. Utterly unbecoming. And which are the soft or drinking harmonies? The Ionian, he replied, and the Lydian; they are termed ‘relaxed.’ Well, and are these of any military use? Quite the reverse, he replied; and if so the Dorian and the Phrygian are the only ones which you have left. I answered: Of the harmonies I know nothing, but I want to have one warlike, to sound the note or accent which a brave man utters in the hour of danger and stern resolve, or when his cause is failing, and he is going to wounds or death or is overtaken by some other evil, and at every such crisis meets the blows of fortune with firm step and a determination to endure; and another to be used by him in times of peace and freedom of action, when there is no pressure of necessity, and he is seeking to persuade God by prayer, or man by instruction and admonition, or on the other hand, when he is expressing his willingness to yield to persuasion or entreaty or admonition, and which represents him when by prudent conduct he has attained his end, not carried away by his success, but acting moderately and wisely

(1) Pythagoras was likewise of opinion that music contributed greatly to health, if it was used in an appropriate manner. For he was accustomed to employ a purification of this kind, but not in a careless way. And he called the medicine which is obtained through music by the name of purification. But he employed such a melody as this about the vernal season. For he placed in the middle a certain person who played on the lyre, and seated in a circle round him those who were able to sing. And thus, when the person in the centre struck the lyre, those that surrounded him sung certain pæans, through which they were seen to be delighted, and to become elegant and orderly in their manners.

But at another time they used music in the place of medicine. And there are certain melodies devised as remedies against the passions of the soul, and also against despondency and lamentation, which Pythagoras invented as things that afford the greatest assistance in these maladies. And again, he employed other melodies against rage and anger, and against every aberration of the soul. There is also another kind of modulation invented as a remedy against desires. He likewise used dancing; but employed the lyre as an instrument for this purpose. For he conceived that the pipe was calculated to excite insolence, was a theatrical instrument, and had by no means a liberal sound. Select verses also of Homer and Hesiod were used by him, for the purpose of correcting the soul.

Among the deeds of Pythagoras likewise, it is said, that once through the spondaic song of a piper, he extinguished the rage of a Tauromenian lad, who had been feasting by night, and intended to burn the vestibule of his mistress, in consequence of seeing her coming from the house of his rival. For the lad was inflamed and excited [to this rash attempt] by a Phrygian song; which however Pythagoras most rapidly suppressed. But Pythagoras, as he was astronomizing, happened to meet with the Phrygian piper at an unseasonable time of night, and persuaded him to change his Phrygian for a spondaic song; through which the fury of the lad being immediately repressed, he returned home in an orderly manner, though a little before this, he could not be in the least restrained, nor would in short, bear any admonition; and even stupidly insulted Pythagoras when he met him.

When a certain youth also rushed with a drawn sword on Anchitus, the host of Empedocles, because, being a judge, he had publicly condemned his father to death, and would have slain him as a homicide, Empedocles changed the intention of the youth, by singing to his lyre that verse of Homer,

(1) Conceiving, however, that the first attention which should be paid to men, is that which takes place through the senses; as when some one perceives beautiful figures and forms, or hears beautiful rythms and melodies, he established that to be the first erudition which subsists through music, and also through certain melodies and rythms, from which the remedies of human manners and passions are obtained, together with those harmonies of the powers of the soul which it possessed from the first. He likewise devised medicines calculated to repress and expel the diseases both of bodies and souls. And by Jupiter that which deserves to be mentioned above all these particulars is this, that he arranged and adapted for his disciples what are called apparatus and contrectations, divinely contriving mixtures of certain diatonic, chromatic, and euharmonic melodies, through which he easily transferred and circularly led the passions of the soul into a contrary direction, when they had recently and in an irrational and clandestine manner been formed; such as sorrow, rage, and pity, absurd emulation and fear, all-various desires, angers, and appetites, pride, supineness, and vehemence.

For he corrected each of these by the rule of virtue, attempering them through appropriate melodies, as through certain salutary medicines. In the evening, likewise, when his disciples were retiring to sleep, he liberated them by these means from diurnal perturbations and tumults, and purified their intellective power from the influxive and effluxive waves of a corporeal nature; rendered their sleep quiet, and their dreams pleasing and prophetic. But when they again rose from their bed, he freed them from nocturnal heaviness, relaxation and torpor, through certain peculiar songs and modulations, produced either by simply striking the lyre, or employing the voice. Pythagoras, however, did not procure for himself a thing of this kind through instruments or the voice, but employing a certain ineffable divinity, and which it is difficult to apprehend, he extended his ears, and fixed his intellect in the sublime symphonies of the world, he alone hearing and understanding, as it appears, the universal harmony and consonance of the spheres, and the stars that are moved through them, and which produce a fuller and more intense melody than any thing effected by mortal sounds.

This melody also was the result of dissimilar and variously differing sounds, celerities, magnitudes, and intervals, arranged with reference to each other in a certain most musical ratio, and thus producing a most gentle, and at the same time variously beautiful motion and convolution. Being therefore irrigated as it were with this melody, having the reason of his intellect well arranged through it, and as I may say, exercised, he determined to exhibit certain images of these things to his disciples as much as possible, especially producing an imitation of them through instruments, and through the mere voice alone. For he conceived that by him alone, of all the inhabitants of the earth, the mundane sounds were understood and heard, and this from a natural fountain itself and root.

He therefore thought himself worthy to be taught, and to learn something about the celestial orbs, and to be assimilated to them by desire and imitation, as being the only one on the earth adapted to this by the conformation of his body, through the dæmoniacal power that inspired him. But he apprehended that other men ought to be satisfied in looking to him, and the gifts he possessed, and in being benefited and corrected through images and examples, in consequence of their inability to comprehend truly the first and genuine archetypes of things. Just, indeed, as to those who are incapable of looking intently at the sun, through the transcendent splendor of his rays, we contrive to exhibit the eclipses of that luminary, either in the profundity of still water, or through melted pitch, or through some darkly-splendid mirror; sparing the imbecility of their eyes, and devising a method of representing a certain repercussive light, though less intense than its archetype, to those who are delighted with a thing of this kind. Empedocles also appears to have obscurely signified this about Pythagoras, and the illustrious and divinely-gifted conformation of his body above that of other men, when he says:

(4) While the early Chinese, Hindus, Persians, Egyptians, Israelites, and Greeks employed both vocal and instrumental music in their religious ceremonials, also to complement their poetry and drama, it remained for Pythagoras to raise the art to its true dignity by demonstrating its mathematical foundation. Although it is said that he himself was not a musician, Pythagoras is now generally credited with the discovery of the diatonic scale. Having first learned the divine theory of music from the priests of the various Mysteries into which he had been accepted, Pythagoras pondered for several years upon the laws governing consonance and dissonance. How he actually solved the problem is unknown, but the following explanation has been invented.

(1) What you afterwards say is as follows: “ That some of those who suffer a mental alienation, energize enthusiastically on hearing cymbals or drums, or a certain modulated sound, such as those who are Corybantically inspired, those who are possessed by Sabazius, and those who are inspired by the mother of the Gods .” It is necessary, therefore, to discuss the causes of these things, and to show how they are definitely produced. That music, therefore, is of a motive nature, and is adapted to excite the affections, and that the melody of pipes produces or heals the disordered passions of the soul, changes the temperaments or dispositions of the body, and by some melodies causes a Bacchic fury, but by others occasions this fury to cease; and, likewise, how the differences of these accord with the several dispositions of the soul, and that an unstable and variable melody is adapted to ecstasies, such as are the melodies of Olympus, and others of the like kind; all these appear to me to be adduced in a way foreign to enthusiasm. For they are physical and human, and the work of our art; but nothing whatever of a divine nature in them presents itself to the view.

(6) Pythagoras thereupon discovered that the first and fourth strings when sounded together produced the harmonic interval of the octave, for doubling the weight had the same effect as halving the string. The tension of the first string being twice that of the fourth string, their ratio was said to be 2:1, or duple. By similar experimentation he ascertained that the first and third string produced the harmony of the diapente, or the interval of the fifth. The tension of the first string being half again as much as that of the third string, their ratio was said to be 3:2, or sesquialter. Likewise the second and fourth strings, having the same ratio as the first and third strings, yielded a diapente harmony. Continuing his investigation, Pythagoras discovered that the first and second strings produced the harmony of the diatessaron, or the interval of the third; and the tension of the first string being a third greater than that of the second string, their ratio was said to be 4:3, or sesquitercian. The third and fourth strings, having the same ratio as the first and second strings, produced another harmony of the diatessaron. According to Iamblichus, the second and third strings had the ratio of 8:9, or epogdoan.

(3) It is highly probable that the Greek initiates gained their knowledge of the philosophic and therapeutic aspects of music from the Egyptians, who, in turn, considered Hermes the founder of the art. According to one legend, this god constructed the first lyre by stretching strings across the concavity of a turtle shell. Both Isis and Osiris were patrons of music and poetry. Plato, in describing the antiquity of these arts among the Egyptians, declared that songs and poetry had existed in Egypt for at least ten thousand years, and that these were of such an exalted and inspiring nature that only gods or godlike men could have composed them. In the Mysteries the lyre was regarded as the secret symbol of the human constitution, the body of the instrument representing the physical form, the strings the nerves, and the musician the spirit. Playing upon the nerves, the spirit thus created the harmonies of normal functioning, which, however, became discords if the nature of man were defiled.

(25) The names given by the Pythagoreans to the various notes of the diatonic scale were, according to Macrobius, derived from an estimation of the velocity and magnitude of the planetary bodies. Each of these gigantic spheres as it rushed endlessly through space was believed to sound a certain tone caused by its continuous displacement of the æthereal diffusion. As these tones were a manifestation of divine order and motion, it must necessarily follow that they partook of the harmony of their own source. "The assertion that the planets in their revolutions round the earth uttered certain sounds differing according to their respective 'magnitude, celerity and local distance,' was commonly made by the Greeks. Thus Saturn, the farthest planet, was said to give the gravest note, while the Moon, which is the nearest, gave the sharpest. 'These sounds of the seven planets, and the sphere of the fixed stars, together with that above us [Antichthon], are the nine Muses, and their joint symphony is called Mnemosyne.'" (See The Canon.)This quotation contains an obscure reference to the ninefold division of the universe previously mentioned.

(28) The Pythagoreans believed that everything which existed had a voice and that all creatures were eternally singing the praise of the Creator. Man fails to hear these divine melodies because his soul is enmeshed in the illusion of material existence. When he liberates himself from the bondage of the lower world with its sense limitations, the music of the spheres will again be audible as it was in the Golden Age. Harmony recognizes harmony, and when the human soul regains its true estate it will not only hear the celestial choir but also join with it in an everlasting anthem of praise to that Eternal Good controlling the infinite number of parts and conditions of Being.

(2) Again, however, assuming a more elevated exordium, I am desirous to exhibit the principles of the worship of the Gods, which Pythagoras and his followers established; viz. that all such particulars as they define with respect to doing or not doing a thing, have for the mark at which they aim, a consent with divinity. This also is with them the principle, [of piety] and their whole life is arranged with a view to follow God . The language, too, of their philosophy is this, that men act ridiculously in exploring good from any other source than the Gods; and that their conduct in this respect resembles that of a man, who in a country governed by a king should reverence one of the magistrates in the city, and neglect him who is the ruler of all of them.

For they were of opinion that such was the conduct of mankind. For since God is, and is the Lord of all things, it is universally acknowledged that good is to be requested of him. For all men impart good to those whom they love, and to those with whom they are delighted; but they give the contrary to good, to those to whom they are contrarily disposed. It is evident, therefore, that those things are to be done, in which God delights. It is, however, not easy for a man to know what these are, unless he obtains this knowledge from one who has heard God, or has heard God himself, or procures it through divine art. Hence also, the Pythagoreans were studious of divination.

For this alone is an interpretation of the benevolence of the Gods. And in short, he will conceive an employment of this kind to be worthy of regard, who believes that there are Gods; but he who thinks that either of these is folly, will also be of opinion that both are foolish. Many of the mandates, however, of the Pythagoreans were introduced from the mysteries; for they did not conceive them to be the productions of arrogance, but to originate from a certain divinity. And in a similar manner, all the Pythagoreans believe such things as are mythologically related of Aristeas the Proconesian, and Abaris the Hyperborean, and other particulars of a like nature. For they consider every thing of this kind to be credible; and of many [such] things they make trial themselves.

They also frequently recollect such-like particulars as appear to be fabulous, as not disbelieving in any thing which may be referred to divinity. A certain person therefore relates, that Eurytus said, that a shepherd feeding his sheep near the tomb of Philolaus, heard some one singing. But the person to whom this was related, did not at all disbelieve the narration, but asked what kind of harmony it was? Both of them, however, were Pythagoreans, and Eurytus was the disciple of Philolaus. It is likewise said, that a certain person told Pythagoras, that he appeared to himself once to converse with his father who was dead, and that he asked Pythagoras what this indicated? Pythagoras replied, that it indicated nothing; but that he had in reality conversed with his father.

As therefore, said he, nothing is signified by my now discoursing with you, so neither is any thing signified by your conversing with your father. Hence, in all particulars of this kind, they did not think that they were stupid, but those that disbelieved in them. For they did not conceive that some things are possible to the Gods, but others impossible, as those fancy who reason sophistically; but they believed that all things are possible to the Gods. And this very assertion is the beginning of the verses, which they ascribe to Linus, and which are as follow:

(2) Nor is it without cause the Muses’ choir hath been sent down by Highest Deity unto the host of men; in order that, forsooth, the terrene world should not seem too uncultured, had it lacked the charm of measures, but rather that with songs and praise of men accompanied with music, He might be lauded,—He who alone is all, or is the Sire of all; and so not even on the earths, should there have been an absence of the sweetness of the harmony of heavenly praise.

(400) a courageous and harmonious life; and when we have found them, we shall adapt the foot and the melody to words having a like spirit, not the words to the foot and melody. To say what these rhythms are will be your duty—you must teach me them, as you have already taught me the harmonies. But, indeed, he replied, I cannot tell you. I only know that there are some three principles of rhythm out of which metrical systems are framed, just as in sounds there are four notes 36 out of which all the harmonies are composed; that is an observation which I have made. But of what sort of lives they are severally the imitations I am unable to say. Then, I said, we must take Damon into our counsels; and he will tell us what rhythms are expressive of meanness, or insolence, or fury, or other unworthiness, and what are to be reserved for the expression of opposite feelings. And I think that I have an indistinct recollection of his mentioning a complex Cretic rhythm; also a dactylic or heroic, and he arranged them in some manner which I do not quite understand, making the rhythms equal in the rise and fall of the foot, long and short alternating; and, unless I am mistaken, he spoke of an iambic as well as of a trochaic rhythm, and assigned to them short and long quantities. 37 Also in some cases he appeared to praise or censure the movement of the foot quite as much as the rhythm; or perhaps a combination of the two; for I am not certain what he meant. These matters, however, as I was saying, had better be referred to Damon himself, for the analysis of the subject would be difficult, you know? Rather so, I should say. But there is no difficulty in seeing that grace or the absence of grace is an effect of good or bad rhythm. None at all.

(11) To Pythagoras music was one of the dependencies of the divine science of mathematics, and its harmonies were inflexibly controlled by mathematical proportions. The Pythagoreans averred that mathematics demonstrated the exact method by which the good established and maintained its universe. Number therefore preceded harmony, since it was the immutable law that governs all harmonic proportions. After discovering these harmonic ratios, Pythagoras gradually initiated his disciples into this, the supreme arcanum of his Mysteries. He divided the multitudinous parts of creation into a vast number of planes or spheres, to each of which he assigned a tone, a harmonic interval, a number, a name, a color, and a form. He then proceeded to prove the accuracy of his deductions by demonstrating them upon the different planes of intelligence and substance ranging from the most abstract logical premise to the most concrete geometrical solid. From the common agreement of these diversified methods of proof he established the indisputable existence of certain natural laws.

(5) One day while meditating upon the problem of harmony, Pythagoras chanced to pass a brazier's shop where workmen were pounding out a piece of metal upon an anvil. By noting the variances in pitch between the sounds made by large hammers and those made by smaller implements, and carefully estimating the harmonies and discords resulting from combinations of these sounds, he gained his first clue to the musical intervals of the diatonic scale. He entered the shop, and after carefully examining the tools and making mental note of their weights, returned to his own house and constructed an arm of wood so that it: extended out from the wall of his room. At regular intervals along this arm he attached four cords, all of like composition, size, and weight. To the first of these he attached a twelve-pound weight, to the second a nine-pound weight, to the third an eight-pound weight, and to the fourth a six-pound weight. These different weights corresponded to the sizes of the braziers' hammers.