ADDITIONAL NOTES. · 3

P. 107. For Pythagoras always proclaimed, that nothing admirable pertaining to the Gods or divine dogmas, should be disbelieved.

This in the Protreptics forms the fourth symbol, and is thus explained by Iamblichus:—“This dogma sufficiently venerates and unfolds the transcendency of the Gods, affording us a viaticum, and recalling to our memory that we ought not to estimate divine power from our judgment. But it is likely that some things should appear difficult and impossible to us, in consequence of our corporeal subsistence, 319 and from our being conversant with generation and corruption; from our having a momentary existence; from being subject to a variety of diseases; from the smallness of our habitation; from our gravitating tendency to the middle; from our somnolency, indigence and repletion; from our want of counsel and our imbecility; from the impediments of our soul, and a variety of other circumstances, although our nature possesses many illustrious prerogatives. At the same time however we perfectly fall short of the Gods, and neither possess the same power with them, nor equal virtue. This symbol therefore in a particular manner introduces the knowledge of the Gods, as beings who are able to effect all things. On this account it exhorts us to disbelieve nothing concerning the Gods. It also adds, nor about divine dogmas; viz. those belonging to the Pythagoric philosophy. For these being secured by disciplines and scientific theory, are alone true and free from falsehood, being corroborated by all-various demonstration, accompanied with necessity. The same symbol, also, is capable of exhorting us to the science concerning the Gods: for it urges us to acquire a science of that kind, through which we shall be in no respect deficient in things asserted about the Gods. It is also able to exhort the same things concerning divine dogmas, and a disciplinative progression. For disciplines alone give eyes to, and produce light about, all 320 things, in him who intends to consider and survey them. For from the participation of disciplines, one thing before all others is effected, viz. a belief in the nature, essence, and power of the Gods, and also in those Pythagoric dogmas, which appear to be prodigious to such as have not been introduced to, and are uninitiated in, disciplines; So that the precept disbelieve not is equivalent to participate and acquire those things through which you will not disbelieve; that is to say, acquire disciplines and scientific demonstrations.”

P. 88. After this manner therefore it is said that music was discovered by Pythagoras.

The following particulars relative to music are added for the purpose of elucidating what is said about it in this chapter.

“Take two brazen chords, such as are used in harps; for those chords which are made from the intestines of sheep are for the most part either false or obnoxious to the change of the air,

A—————————B C—————————D | E

“Let these chords be perfectly equal, and equally 321 stretched, so as to be in unison, i. e. so that there may be only one sound, though there are two strings. But it is requisite that they should be placed upon some oblong and polished rule. The ancients called this rule an harmonic rule, or also a monochord, by which instrument all consonances and dissonances, and likewise musical intervals, were tried. Let now one of these chords be bisected in E. Afterwards under the point E place what is vulgarly called the tactus, but which was denominated by the ancients, from its figure, a hemisphere. The tactus, therefore, being placed under E, press there the chord, so that one half of it only, as for instance ED, may be wholly struck and resound. Having therefore struck each of the chords at the same time, viz. the whole of AB, and the half ED, so that they may resound at one and the same time, you will hear the sweetest of all consonances, composed from the sound of the whole chord AB, and the sound of the half ED. This consonance the ancients called diapason, i. e. through all [the chords], because in the musical instruments of the ancients, the two extreme chords, i. e. the most grave, and the most acute of all the chords, contained this consonance; so that, from the gravest chord having made a transition through all the chords to the supreme and most acute of all, they would hear this sweetest consonance. It was, likewise, said to be in a duple ratio of the proportion 322 of one sound to the other. For the sound of the chord AB is doubly greater or more grave than the sound of the half ED. For as sounding bodies are to each other, so are their sounds. But the chord AB is the double of ED. This, however, is now commonly called the octave, because from the first sound, and that the gravest, which is called ut, as far as to that sound which corresponds to it in the consonance diapason, there are these eight sounds, ut, re, mi, fa, sol, re, mi, fa. And of these the first ut, and the last fa, which is the eighth, produce the consonance diapason, or the double, or the octave.

“Again, let the same chord CD be divided into three equal parts in the points F, G.

A————————————B C————————————D | | F G

“FD, therefore, will be two-thirds as well of the whole CD as of the whole AB. Let the tactus now be placed in F, and let AB and FD be struck at the same time, and a consonance very sweet and perfect will indeed be heard, yet not so sweet as the diapason. This the ancients called diapente (i. e. through five chords), because the first and the fifth chord produce this consonance. But according 323 to proportion it is called sesquialter, because the chord AB is sesquialter to FD, and consequently the sounds of these chords also are in the same ratio. But sesquialter ratio is when the greater quantity AB contains the less FD once, and the half of it besides. It is, indeed, commonly called the fifth, because it is composed from the first sound ut, and the fifth, sol.

“Again, let the same chord be cut into four equal parts in the points H, E, I,

A——————————————————————————B C——————————————————————————D | | | | | | | | | K L H F M N E G I

“so that the chord HD, may be three-fourths of the whole CD. The tactus, therefore, being placed in H, let AB and HD be struck at one and the same time, and a consonance will be heard, indeed, yet more imperfect than the preceding two. This was called by the ancients diatessaron, i. e. through four chords or sounds, for a similar reason to that by which the former were denominated. With reference, however, to the ratio of the chords and sounds, it is called sesquitertian, because the greater AB contains the less once, and a third part of it besides. But it is now commonly called a 324 fourth, because it is found between the first sound ut, and the fourth fa. If now the point F be added in the preceding figure, and at one and the same time two chords HD and FD are compared in arithmetical ratios, we shall find that the greater HD will have to the less FD a sesquioctave[105] ratio, and the sound of the greater HD to the less FD will have the same ratio, i. e. in modern terms, that between fa and sol there is a sesquioctave ratio. But if these two sounds are heard together, they will be discordant to the ear. Again, the distance between these sounds fa, sol, or between the chords HD and FD, or between the two harmonic intervals HD and FD, the ratio of which was sesquioctave, was called by the ancients a tone. Afterwards they divided the whole of CD into nine equal parts, the first of which is divided in K, so that the whole CD may have to the remainder KD, which contains eight of those parts, a sesquioctave ratio. This, in like manner, will be the interval of a tone, the first sound of which, i. e, of the whole CD, is now called ut, but the second sound of the rest of the chord KD is called re. Afterwards they in a similar manner divided the remainder KD into nine parts, the first part of which is marked in the point L. And for the same reason between the chord KD and the chord KD, 325 and their sounds, there will be a sesquioctave ratio. The sound of the chord LD is now called mi; but the interval which remains between the chord LD and the chord HD has not a sesquioctave ratio, but less than it almost by half, and therefore an interval of this kind was called a semitone, and also diesis or a division. But that interval which remains between the points F and E they divided after the same manner, as the space between C and H was divided, and they again found the same sounds. Let those divisions be marked by the points M and N; and here, also, between N and E, or between mi and fa, there is in like manner another semitone. These eight sounds, therefore, are ut, re, mi, fa, sol, re, mi, fa, which compose the whole diapason. For as we have before observed, between ut and the last fa is the consonance diapason, or between the chord CD or AB, and the chord ED. But from the intervals which are between the sounds there are two semitones, viz. one between mi and fa, denoted by the letters L, N, and the other between the last mi and fa, denoted by the letters N, E. The remaining five intervals are entire tones. It must, also, be observed, that from ut to the first sol is the consonance diapente, which contains three tonic intervals, and one semitone; nevertheless in all there are five sounds, ut, re, mi, fa, sol.

“Again, from sol to the last fa there are four 326 sounds, sol, re, mi, fa, which are perfectly similar to the first four, ut, re, mi, fa. Nevertheless these are more grave, but those are more acute. And as from ut to the first fa is the diatessaron, so likewise from sol to the last fa is another diatessaron, from which, in the last place, it must be observed, it follows that the two consonances diatessaron and diapente constitute the whole diapason; or that the diapason is divided into one diatessaron, and one diapente. For from ut to sol is the diapente, but from sol to the last fa is the diatessaron. This will also be the case if we should say that from ut to the first fa is the diatessaron, as is evident from the division of the chord; but from the first fa to the last fa is the diapente, as is evident from the four intervals of the chord, three of which are tones, and the remaining interval is a semitone, which also in the other diapente were contained between ut and sol.

“Now again, let the tactus be placed in I; but I is the fourth part of the whole CD. Let, also, AB and ID be struck at one and the same time, and the sweetest consonance, called bisdiapason, will be produced; which is so denominated, because it is composed from two diapasons, of which the first is between AB or CD, and ED, but the second is between ED and ID; for the ratio of these is double as well as of those. The ratio, also, of the bisdiapason is quadruple, as is evident from 327 the division; and is commonly called a fifteenth, because from the first ut to this sound, which is also denominated fa, there would be fifteen sounds, if the interval EI were divided after the same manner as the first CE is divided.

“Farther still, let GD be a third part of the whole CD, and let the tactus be placed in G. Then at one and the same time let AB and GD be struck, and a sweet consonance will be heard, which is called diapasondiapente, because it is composed from one diapason contained by the interval CE, or the two chords CD, ED, and one diapente, contained by the interval EG, or the chords ED, GD. For the chord ED is sesquialter to the chord GD; which ratio constitutes the nature of the diapente. The proportion, also, of this consonance is triple. For the chord AB or CD is triple of GD; and it is commonly called the twelfth, because between ut and sol, denoted by the letter G, there would be twelve sounds, if the interval EG received its divisions. From all which it is manifest by the experience of the ear, that there are altogether five consonances, three simple, the diapason, the diapente, and the diatessaron; but two composite, the bisdiapason, and the diapasondiapente.”

In the last place, it is necessary to observe that those ancient Greeks differently denominated these sounds, ut, re, &c. For the first, i. e. the gravest 328 sound or chord, which is now called ut, they, denominated hypate, and the others in the following order:

P. 109. I swear by him who the tetractys found.

The tetrad was called by the Pythagoreans every number, because it comprehends in itself all the numbers as far as to the decad, and the decad itself; for the sum of 1, 2, 3, and 4, is 10. Hence both the decad and the tetrad were said by them to be every number; the decad indeed in energy, but the tetrad in capacity. The sum likewise of these four numbers was said by them to constitute the tetractys, in which all harmonic ratios are included. For 4 to 1, which is a quadruple ratio, forms the symphony bisdiapason; the ratio of 3 to 2, which is sesquialter, forms the symphony diapente; 4 to 329 3, which is sesquitertian, the symphony diatessaron; and 2 to 1, which is a duple ratio, forms the diapason.

In consequence, however, of the great veneration paid to the tetractys by the Pythagoreans, it will be proper to give it a more ample discussion, and for this purpose to show from Theo of Smyrna,[106] how many tetractys there are: “The tetractys,” says he, “was not only principally honored by the Pythagoreans, because all symphonies are found to exist within it, but also because it appears to contain the nature of all things.” Hence the following was their oath: “Not by him who delivered to our soul the tetractys, which contains the fountain and root of everlasting nature.” But by him who delivered the tetractys they mean Pythagoras; for the doctrine concerning it appears to have been his invention. The above-mentioned tetractys, therefore, is seen in the composition of the first numbers 1. 2. 3. 4. But the second tetractys arises from the increase by multiplication of even and odd numbers beginning from the monad.

Of these, the monad is assumed as the first, because, as we have before observed, it is the principle of all even, odd, and evenly-odd numbers, and the nature of it is simple. But the three successive numbers receive their composition according to 330 the even and the odd; because every number is not alone even, nor alone odd. Hence the even and the odd receive two tetractys, according to multiplication; the even indeed, in a duple ratio; for 2 is the first of even numbers, and increases from the monad by duplication. But the odd number is increased in a triple ratio; for 3 is the first of odd numbers, and is itself increased from the monad by triplication. Hence the monad is common to both these, being itself even and odd. The second number, however, in even and double numbers is 2; but in odd and triple numbers 3. The third among even numbers is 4; but among odd numbers is 9. And the fourth among even numbers is 8; but among odd numbers is 27.

{ 1. 2. 4. 8. } { 1. 3. 9. 27. }

In these numbers the more perfect ratios of symphonies are found; and in these also a tone is comprehended. The monad, however, contains the productive principle of a point. But the second numbers 2 and 3 contain the principle of a side, since they are incomposite, and first, are measured by the monad, and naturally measure a right line. The third terms are 4 and 9, which are in power a square superficies, since they are equally equal. And the fourth terms 8 and 27 being equally equally equal, are in power a cube. Hence from these 331 numbers, and this tetractys, the increase takes place from a point to a solid. For a side follows after a point, a superficies after a side, and a solid after a superficies. In these numbers also, Plato in the Timæus constitutes the soul. But the last of these seven numbers, i. e. 27, is equal to all the numbers that precede it; for 1 + 2 + 3 + 4 + 8 + 9 = 27. There are, therefore, two tetractys of numbers, one of which subsists by addition, but the other by multiplication, and they comprehend musical, geometrical, and arithmetical ratios, from which also the harmony of the universe consists.

But the third tetractys is that which according to the same analogy or proportion comprehends the nature of all magnitude. For what the monad was in the former tetractys, that a point is in this. What the numbers 2 and 3, which are in power a side, were in the former tetractys, that the extended species of a line, the circular and the right, are in this; the right line indeed subsisting in conformity to the even number, since it is terminated[107] by two points; but the circular in conformity to the odd number, because it is comprehended by one line which has no end. But what in the former tetractys the square numbers 4 and 9 were, that the two-fold 332 species of planes, the rectilinear and the circular, are in this. And what the cube numbers 8 and 27 were in the former, the one being an even, but the other an odd number, that the two solids, one of which has a hollow superficies, as the sphere and the cylinder, but the other a plane superficies, as the cube and pyramid, are in this tetractys. Hence, this is the third tetractys, which gives completion to every magnitude, from a point, a line, a superficies, and a solid.

The fourth tetractys is of the simple bodies fire, air, water, and earth, which have an analogy according to numbers. For what the monad was in the first tetractys, that fire is in this. But the duad is air, the triad is water, and the tetrad is earth. For such is the nature of the elements according to tenuity and density of parts. Hence fire has to air the ratio of 1 to 2; but to water, the ratio of 1 to 3; and to earth, the ratio of 1 to 4. In other respects also they are analogous to each other.

The fifth tetractys is of the figures of the simple bodies. For the pyramid, indeed, is the figure of fire; the octaedron, of air; the icosaedron, of water; and the cube, of earth.

The sixth tetractys is of things rising into existence through the vegetative life. And the seed, indeed, is analogous to the monad and a point. 333 But if it increases in length it is analogous to the duad and a line; if in breadth, to the triad and a superficies; but if in thickness, to the tetrad and a solid.

The seventh tetractys is of communities; of which the principle indeed, and as it were monad, is man; the duad is a house; the triad a street; and the tetrad a city. For a nation consists of these. And these indeed are the material and sensible tetractys.

The eighth tetractys consists of the powers which form a judgment of things material and sensible, and which are of a certain intelligible nature. And these are, intellect, science, opinion, and sense. And intellect, indeed, corresponds in its essence to the monad; but science to the duad; for science is the science of a certain thing. Opinion subsists between science and ignorance; but sense is as the tetrad. For the touch which is common to all the senses being fourfold, all the senses energize according to contact.

The ninth tetractys is that from which the animal is composed, the soul and the body. For the parts of the soul, indeed, are the rational, the irascible, and the epithymetic, or that which desires external good; and the fourth is the body in which the soul subsists.