περὶ δὲ φύσιος καὶ ἁρμονίας ὧδε ἔχει· ἁ μὲν ἐστὼ τῶν πραγμάτων ἀίδιος ἔσσα καὶ αὐτὰ μὲν ἁ φύσις θείαν γα καὶ οὐκ ἀνθρωπίνην ἐνδέχεται γνῶσιν πλέον γα ἢ ὅτι οὐχ οἷόν τ’ ἦν οὐθὲν τῶν ἐόντων καὶ γιγνωσκόμενον ὑφ’ ἁμῶν γα γενέσθαι μὴ ὑπαρχούσας τᾶς ἐκτοῦς τῶν πρα γμάτων, ἐξ ὧν συνέστα ὁ κόσμος, καὶ τῶν περαινόντων καὶ τῶν ἀπείρων. ἐπεὶ δὲ ταὶ ἀρχαὶ ὑπᾶρχον οὐχ ὁμοῖαι οὐδ’ ὁμόφυλοι ἔσσαι, ἢδη ἀδύνατον ἧς κα αὐταῖς κοσμηθῆναι, εἰ μὴ ἁρμονία ἐπεγένετο ὡιτινιῶν ἅδε τρόπωι ἐγένετο. τὰ μὲν ὦν ὁμοῖα καὶ ὁμόφυλα ἁρμονίας οὐδὲν ἐπεδέοντο, τὰ δὲ ἀνόμοια μηδὲ ὁμόφυλα μηδὲ ἰσοταγῆ ἀνάγκα τᾶι τοιαύται ἁρμονίαι συγκεκλεῖσθαι, οἵαι μέλλοντι ἐν κόσμωι κατέχεσθαι. ἁρμονίας δὲ μέγεθός ἐστι συλλαβὰ καὶ δι’ ὀξειᾶν· τὸ δὲ δι’ ὀξειᾶν μεῖζον τᾶς συλλαβᾶς ἐπογδόωι. ἔστι γὰρ ἀπὸ ὑπάτας ἐπὶ μέσσαν συλλαβά, ἀπὸ δὲ μέσσας ἐπὶ νεάταν δι’ ὀξειᾶν, ἀπὸ δὲ νεάτας ἐς τρίταν συλλαβά, ἀπὸ δὲ τρίτας ἐς ὑπάταν δι’ ὀξειᾶν· τὸ δ’ ἐν μέσωι μέσσας καὶ τρίτας ἐπόγδοον· ἁ δὲ συλλαβὰ ἐπίτριτον, τὸ δὲ δι’ ὀξειᾶν ἡμιόλιον, τὸ διὰ πασᾶν δὲ διπλόον. οὕτως ἁρμονία πέντε ἐπόγδοα καὶ δύο διέσιες, δι’ ὀξειᾶν δὲ τρία ἐπόγδοα καὶ δίεσις, συλλαβὰ δὲ δύ’ ἐπόγδοα καὶ δίεσις.
Philolaus igitur haee atque his minora spatia talibus definitionibus includit: comma vero est spatium, quo maior est sesquioetaea proportio duabas diesibus, id est duobus semitoniis minoribus. schisma est dimidium eom matis, diaschisma vero dimidium dieseos, id est semitonii minoris.
This is the state of affairs about nature and harmony. The essence of things is eternal; it is a unique and divine nature, the knowledge of which does not belong to man. Still it would not be possible that any of the things that are, and are known by us, should arrive to our knowledge, if this essence was not the internal foundation of the principles of which the world was founded, that is, of the finite and infinite elements. Now since these principles are not mutually similar, neither of similar nature, it would be impossible that the order of the world should have been formed by them, unless the harmony had intervened, in any manner whatever. Of course, the things that were similar, and of similar nature, did not need the harmony; but the dissimilar things, which have neither a similar nature, nor an equivalent function, must be organized by the harmony, if they are to take their place in the connected totality of the world.
The extent of the harmony is a fourth, plus a fifth. The fifth is greater than the fourth by nine eighths; for from the lowest string to the second lowest, a fourth; and from this to the next, a fifth; but from this to the next, or "third," a fourth; and from this "third" to the lowest, a fifth. The interval between the second lowest and the "third" (from the top) is nine eighths; the interval of the fourth, is four thirds; that of the fifth, three halves; that of the octave, the double relation. Thus the harmony contains five nine-eighths plus two sharps; the fifth, three nine eighths, plus one sharp; the fourth two nine-eighths, plus one.
These are the definitions that Philolaus has given of these intervals, and of still smaller intervals. The comma, says he, is the interval whose nine eighths relation exceeds the sum of two sharps, namely, the sum of two minor semi-tones. The schisma is half the comma, the diaschisma is half the sharp, namely, of the minor semi-tone.
Kenneth Sylvan Guthrie · Pythagorean Library (1920)