§19.41Διὰ τί δὶς μὲν δι’ ὀξειῶν ἢ δὶς διὰ τεττάρων οὐ συμφωνεῖ, δὶς διὰ πασῶν δέ;
Why is it that a double fifth or a double fourth does not form a concord, but a double octave does?
ἢ ὅτι τὸ μὲν διὰ πέντε ἐστὶν ἐν ἡμιολίῳ λόγῳ, τὸ δὲ διὰ τεττάρων ἐν ἐπιτρίτῳ;
Or is it because the fifth is in a ratio of 3:2, and the fourth is in a ratio of 4:3?
ὄντων δὲ ἡμιολίων δυοῖν ἑξῆς ἀριθμῶν ἢ ἐπιτρίτων οἱ ἄκροι πρὸς ἀλλήλους οὐδένα λόγον ἕξουσιν· οὔτε γὰρ ἐπιμόριοι οὔτε πολλαπλάσιοι ἔσονται.
But when there are two successive ratios of 3:2 or of 4:3, the extreme terms will have no ratio to one another; for they will be neither superparticular nor multiple.
τὸ δὲ διὰ πασῶν, ἐπειδή ἐστιν ἐν διπλασίῳ λόγῳ.
But the octave is in a double ratio.
δὶς τούτου γινομένου, ἐν τετραπλασίῳ λόγῳ ἂν εἶεν οἱ ἄκροι πρὸς ἀλλήλους.
When this happens twice, the extreme terms would be in a quadruple ratio to one another.
ὥστ’ ἐπεὶ συμφωνία εὔλογον ἐχόντων φθόγγων πρὸς ἀλλήλους ἐστίν, λόγον δὲ οἱ μὲν τὸ δὶς διὰ πασῶν διάλειμμα ἔχοντες πρὸς ἀλλήλους φθόγγοι ἔχουσιν, οἱ δὲ τὸ δὶς διὰ τεττάρων ἢ δὶς διὰ πέντε οὐκ ἔχουσιν, οἱ μὲν δὶς διὰ πασῶν σύμφωνοι εἶεν ἄν, οἱ δ’ ἕτεροι οὔ, διὰ τὰ εἰρημένα.
Therefore, since concord belongs to sounds that have a rational ratio to one another, and the sounds that have the interval of a double octave have a ratio to one another, while those of a double fourth or a double fifth do not, the double octave would be concordant, while the others would not, for the reasons mentioned.
§19.42Διὰ τί, ἐάν τις ψήλας τὴν νήτην ἐπιλάβῃ, ἡ ὑπάτη μόνη δοκεῖ ὑπηχεῖν;
Why is it that, if someone strikes the nete and then stops it, the hypate alone seems to resonate in response?
ἢ ὅτι ἡ νεάτη λήγουσα καὶ μαραινομένη ὑπάτη γίνεται;
Or is it because the nete, when ceasing and dying away, becomes hypate?
σημεῖον δὲ τὸ ἀπὸ τῆς ὑπάτης τὴν νεάτην δύνασθαι ᾄδειν· ὡς γὰρ οὔσης αὐτῆς ᾠδῆς νεάτης, τὴν ὁμοιότητα λαμβάνουσιν ἀπ’ αὐτῆς.
A proof of this is that one can sing the nete starting from the hypate; for as though the latter itself were the song of the nete, they derive the similarity from it.
ἐπεὶ δὲ καὶ ἠχὼ ᾠδή τίς ἐστιν, ἁφή ἐστι φωνῆς τῆς νεάτης ληγούσης, ἦχος ὢν ὁ αὐτὸς τῷ τῆς ὑπάτης φθόγγῳ κινεῖ, εἰκότως τῇ ὁμοιότητι τὴν ὑπάτην ἡ νήτη δοκεῖ κινεῖν.
And since echo too is a kind of song, and is a contact with the voice of the nete as it ceases, and being the same sound as the note of the hypate, it moves it, it is reasonable that, through this similarity, the nete seems to move the hypate.
τὴν μὲν γὰρ νεάτην ἴσμεν οὗ κινεῖται ἐπιληφθεῖσα· τὴν δὲ ὑπάτην αὐτὴν ὁρῶντες ἀκατάληπτον οὖσαν, καὶ φθόγγου αὐτῆς ἀκούοντες, ταύτην οἰόμεθα ἠχεῖν·
For we know where the nete is stopped from moving; but seeing the hypate itself untouched, and hearing its sound, we think that this is what is resonating.
ὅπερ ἐπὶ πολλῶν ἡμῖν συμβαίνει, ἐφ’ ὧν μήτε τῷ λογισμῷ μήτε τῇ αἰσθήσει δυνάμεθα εἰδῆσαι τὸ ἀκριβές.
This is what happens to us in many cases where we cannot know the exact truth either by reasoning or by perception.
ἔτι εἰ πληγείσης τῆς νεάτης μάλιστα ἐντεταμένης συμβαίνει τὸν ζυγὸν κινεῖσθαι, οὐθὲν ἂν εἴη θαυμαστόν.
Furthermore, if, when the nete (which is most tightly stretched) is struck, it happens that the yoke moves, it would be nothing surprising.
κινηθέντος δὲ πάσας τὰς χορδὰς συγκινεῖσθαι καί τιν’ ἦχον ποιεῖν οὐκ ἄλογον.
And if, when it moves, all the strings are moved together and make some sound, this is not unreasonable.
ταῖς μὲν οὖν ἄλλαις ὁ τῆς νεάτης φθόγγος ἀλλότριός ἐστιν καὶ λήγων καὶ ἀρχόμενος, τῇ δὲ ὑπάτῃ λήγων ὁ αὐτός.
Now, to the other strings, the sound of the nete is alien both when beginning and when ceasing, but to the hypate, when ceasing, it is the same.
οὗ προστεθέντος τῇ ἰδίᾳ αὐτῆς κινήσει, ἐκείνης δόξαι πάντ’ αὐτὸν εἶναι οὐθὲν ἄτοπον.
When this is added to its own proper movement, it is not at all strange that the whole sound should seem to belong to the latter.
ἔσται δὲ μείζων ἢ ὁ κοινὸς τῶν λοιπῶν χορδῶν ἦχος, ὅτι αἱ μὲν ὑπὸ τῆς νεάτης καθάπερ ὠσθεῖσαι μαλακῶς ἤχησαν, ἡ δὲ νεάτη πάσῃ τῇ αὐτῆς δυνάμει, οὖσα αὐτῶν σφοδροτάτη.
And this sound will be greater than the common sound of the remaining strings, because the others, being as it were softly pushed by the nete, resonated gently, whereas the nete acted with all its power, being the most violent of them.
ὥστε εἰκότως καὶ τὸ δευτερεῖον αὐτῆς κρεῖττον ἂν εἴη ἢ τὸ τῶν ἄλλων· ὥστε καὶ βραχείας κινεῖσθαι, ὡς αὐτῆς τε γεγενημένης.
So it is reasonable that its secondary effect should also be stronger than that of the others, so that even if it moves but slightly, the sound seems to have arisen from itself too.
§19.43Διὰ τί ἥδιον τῆς μονῳδίας ἐστὶν ἐάν τις πρὸς αὐλὸν ἢ λύραν ᾄδη;
Why is it more pleasant if one sings to the accompaniment of a flute or a lyre than monody?
ἢ ὅτι πᾶν τὸ ἥδιον μιχθὲν ἡδίονι ἕν ἐστιν;
Or is it because everything pleasant, when mixed with what is more pleasant, becomes one?
ὁ δὲ αὐλὸς ἡδίων τῆς λύρας, ὥστε καὶ ἡ ᾠδὴ τούτῳ μιχθεῖσα ἢ λύρᾳ ἡδίων ἂν εἴη, ἐπεὶ τὸ μεμιγμένον τοῦ ἀμίκτου ἥδιόν ἐστιν, ἐὰν ἀμφοῖν ἅμα τὴν αἴσθησίν τις λαμβάνῃ.
And the flute is more pleasant than the lyre, so that the song mixed with this would be more pleasant than if mixed with the lyre, since the mixed is more pleasant than the unmixed, if one receives the perception of both at the same time.
οἶνος γὰρ ἡδίων τοῦ ὀξυμέλιτος διὰ τὸ μεμῖχθαι μᾶλλον αὑτοῖς τὰ ὑπὸ τῆς φύσεως μιχθέντα ἢ ὑφ’ ἡμῶν.
For wine is more pleasant than oxymel because things mixed by nature are more thoroughly mixed with one another than those mixed by us.
ἔστι γὰρ καὶ οἶνος ὁ μικτὸς ἐξ ὀξέος καὶ γλυκέος χυμοῦ.
For wine too is a mixture of acid and sweet juice.
δηλοῦσι δὲ καὶ αἱ οἰνώδεις ῥοαὶ καλούμεναι.
This is also shown by the so-called "wine-pomegranates".
ἡ μὲν οὖν ᾠδὴ καὶ ὁ αὐλὸς μίγνυνται αὑτοῖς δι’ ὁμοιότητα (πνεύματι γὰρ ἄμφω γίνεται)· ὁ δὲ τῆς λύρας φθόγγος, ἐπειδὴ οὐ πνεύματι γίνεται ἢ ἧττον αἰσθητὸν ἢ ὁ τῶν αὐλῶν, ἀμικτότερός ἐστι τῇ φωνῇ· ποιῶν δὲ διαφορὰν τῇ αἰσθήσει ἧττον ἡδύνει, καθάπερ ἐπὶ τῶν χυμῶν εἴρηται.
Thus the song and the flute mix with each other because of their similarity (for both are produced by breath); but the sound of the lyre, since it is not produced by breath, or its breath-like quality is less perceptible than that of the flutes, is less mixable with the voice; and by causing a difference to perception, it gives less pleasure, just as has been said in the case of juices.
ἔτι ὁ μὲν αὐλὸς πολλὰ τῷ αὑτοῦ ἤχῳ καὶ τῇ ὁμοιότητι συγκρύπτει τῶν τοῦ ᾠδοῦ ἁμαρτημάτων, οἱ δὲ τῆς λύρας φθόγγοι ὄντες ψιλοὶ καὶ ἀμικτότεροι τῇ φωνῇ, καθ’ ἑαυτοὺς θεωρούμενοι καὶ ὄντες αὐτοῖς συμφανῆ ποιοῦσιν τὴν τῆς ᾠδῆς ἁμαρτίαν, καθάπερ κανόνες ὄντες αὐτῶν.
Furthermore, the flute, by its own sound and similarity, conceals many of the singer's errors; whereas the sounds of the lyre, being bare and less mixable with the voice, when contemplated and existing by themselves, make the error in the song conspicuous, as if they were a standard for them.
πολλῶν δὲ ἐν τῇ ᾠδῇ ἁμαρτανομένων, τὸ κοινὸν ἐξ ἀμφοῖν ἀναγκαῖον χεῖρον γίνεσθαι.
And when many errors are committed in the song, the common result from both must necessarily become worse.