§12Τὸ διὰ πασῶν διάστημά ἐστι διπλάσιον.
The octave interval is double.
Ἐδείξαμεν γὰρ αὐτὸ πολλαπλάσιον.
For we showed that it is multiple.
οὐκοῦν ἤτοι διπλάσιόν ἐστιν — ἢ μεῖζον ἢ διπλάσιον.
Therefore, it is either double, or greater than double.
ἀλλʼ ἐπεὶ ἐδείξαμεν τὸ διπλάσιον διάστημα ἐκ δύο τῶν μεγίστων ἐπιμορίων συγκείμενον, ὥστε εἰ ἔσται τὸ διὰ πασῶν μεῖζον διπλασίου, οὐ συγκείσεται ἐκ δύο μόνων ἐπιμορίων, ἀλλʼ ἐκ πλειόνων, — σύγκειται δὲ ἐκ δύο συμφώνων διαστημάτων, ἐκ τε τοῦ διὰ πέντε καὶ τοῦ διὰ τεσσάρων·
But since we showed that the double interval is composed of the two greatest superparticulars, so that if the octave interval is greater than double, it will not be composed of only two superparticulars, but of more, — and yet it is composed of two consonant intervals, both the fifth interval and the fourth interval; therefore, the octave interval will not be greater than double.
οὐκ ἄρα ἔσται τὸ διὰ πασῶν μεῖζον διπλασίου. διπλάσιον ἄρα.
Therefore, it is double.
Ἀλλʼ ἐπειδὴ τὸ διὰ πασῶν ἐστι διπλάσιον, τὸ δὲ διπλάσιον ἐκ τῶν μεγίστων ἐπιμορίων δύο συνέστηκε, καὶ τὸ διὰ πασῶν ἄρα ἐξ ἡμιολίου καὶ ἐπιτρίτου σννέστηκε, ταῦτα γὰρ μέγιστα.
But since the octave interval is double, and the double is composed of the two greatest superparticulars, the octave interval also is therefore composed of a sesquialter and a sesquitertian, for these are the greatest.
συνέστηκε δὲ ἐκ τοῦ διὰ πέντε καὶ ἐκ τοῦ διὰ τεσσάρων, ὄντων ἐπιμορίων· τὸ μὲν ἄρα διὰ πέντε, ἐπειδὴ μεῖζόν ἐστιν, ἡμιόλιον ἂν εἴη, τὸ δὲ διὰ τεσσάρων ἐπίτριτον.
And it is composed of the fifth interval and the fourth interval, which are superparticulars; therefore, the fifth interval, since it is greater, would be sesquialter, and the fourth interval sesquitertian.
Φανερὸν δὴ, ὅτι καὶ τὸ διὰ πέντε καὶ διὰ πασῶν τριπλάσιόν ἐστιν.
It is therefore clear that the fifth and octave combined is a triple interval.
ἐδείξαμεν γὰρ, ὅτι ἐκ διπλασίου διαστήματος καὶ ἡμιολίου τριπλάσιον διάστημα γίνεται, ὥστε καὶ τὸ διὰ πασῶν καὶ τὸ διὰ πέντε τριπλάσιον.
For we showed that from a double interval and a sesquialter, a triple interval is produced, so that the octave and the fifth combined is triple.
Τὸ δὲ δὶς διὰ πασῶν τετραπλάσιόν ἐστιν.
And the double octave is quadruple.
Ἀποδέδεικται ἄρα τῶν συμφώνων ἕκαστον, ἐν τίσι λόγοις (p. 34) ἔχει τοὺς περιέχοντας φθόγγους πρὸς ἀλλήλους.
Therefore, each of the concordant intervals has been demonstrated, in what ratios (p. 34) it has the sounds that encompass it to one another.
§13Λοιπὸν δὴ περὶ τοῦ τονιαίου διαστήματος διελθεῖν, ὅτι ἐστὶν ἐπόγδοον.
It remains, then, to discuss the whole-tone interval, that it is sesquioctave.
Ἐμάθομεν γὰρ, ὅτι ἐὰν ἀπὸ ἡμιολίου διαστήματος ἐπίτριτον διάστημα ἀφαιρεθῇ, τὸ λοιπὸν καταλείπεται ἐπόγδοον.
For we learned that if a sesquitertian interval is subtracted from a sesquialter interval, the remainder is left as a sesquioctave.
ἐὰν δὲ ἀπὸ τοῦ διὰ πέντε τὸ διὰ τεσσάρων ἀφαιρεθῇ, τὸ λοιπὸν τονιαῖόν ἐστι διάστημα· τὸ ἄρα τονιαῖον διάστημά ἐστιν ἐπόγδοον.
And if the fourth interval is subtracted from the fifth, the remainder is the whole-tone interval; therefore, the whole-tone interval is sesquioctave.
§14Τὸ διὰ πασῶν ἔλαττον ἢ ἓξ τόνων.
The octave interval is less than six whole tones.
Δέδεικται γὰρ τὸ μὲν διὰ πασῶν διπλάσιον, ὁ δὲ τόνος ἐπόγδοος· τὰ δὲ ἓξ ἐπόγδοα διαστήματα μείζονα διαστήματός διπλασίου.
For the octave has been shown to be double, and the whole tone is sesquioctave; and six sesquioctave intervals are greater than a double interval.
τὸ ἄρα διὰ πασῶν ἔλαττόν ἐστιν ἓξ τόνων.
Therefore, the octave interval is less than six whole tones.
§15Τὸ διὰ τεσσάρων ἔλαττον δύο τόνων καὶ ἡμιτονίου, καὶ τὸ διὰ πέντε ἔλαττον τριῶν τόνων καὶ ἡμιτονίου.
The fourth interval is less than two whole tones and a semitone, and the fifth interval is less than three whole tones and a semitone.
Ἔστω γὰρ νήτη μὲν διεζευγμένων ὁ β, παραμέση δὲ ὁ γ, μέση δὲ ὁ δ, (p. 35) ὑπάτη δὲ μέσων ὁ ζ.
For let the nete diezeugmenon be β, let the paramese be γ, and let the mese be δ, (p. 35) and let the hypate meson be ζ.
οὐκοῦν τὸ μὲν γδ δ διάστημα τόνος ἐστὶ, τὸ δὲ βζ, διὰ πασῶν ὂν, ἔλαττον ἓξ τόνων.
Therefore, the γδ interval is a whole tone, and the βζ interval, being an octave, is less than six whole tones.
τὰ λοιπὰ ἄρα, τό τε β καὶ τὸ δ ἶσα ὄντα ἐλάττονά ἐστι πέντε τόνων.
Therefore, the remaining intervals, namely βγ and δζ, being equal, are less than five whole tones.
ὥστε τὸ ἐν τῷ βγ ἔλαττον δύο τόνων καὶ ἡμιτονίου, ὅ ἐστι διὰ τεσσάρων, τὸ δὲ βδ ἔλαττον τριῶν τόνων καὶ ἡμιτονίου, ὅ ἐστι διὰ πέντε.
So that the interval in βγ is less than two whole tones and a semitone, which is a fourth; and that in βδ is less than three whole tones and a semitone, which is a fifth.
§16Ὁ τόνος οὐ διαιρεθήσεται εἰς δύο ἶσα οὔτε εἰς πλείω.
The whole tone will not be divided into two equal parts, nor into more.
Ἐδείχθη γὰρ ὢν ἐπιμόριος· ἐπιμορίου δὲ διαστήματος μέσοι οὔτε πλείους οὔτε εἷς ἀνάλογον ἐμπίπτουσιν.
For it has been shown to be superparticular; and between a superparticular interval, neither several nor even one mean proportional falls.
οὔκ ἄρα διαιρεθήσεται ὁ τόνος εἰς ἶσα.
Therefore, the whole tone will not be divided into equal parts.
§17Αἱ παρανῆται αἱ καὶ λιχανοὶ ληφθήσονται διὰ συμφωνίας οὕτως.
The paranetes, which are also called lichanoi, will be obtained through concord in this way.
Ἔστω γὰρ μέση ὁ β.
For let the mese be β.
ἐπιτετάσθω (p. 36) διὰ τεσσάρων ἐπὶ τὸ γ, καὶ ἀπὸ τοῦ ἀνείσθω διὰ πέντε ἐπὶ τὸ δ.
Let it be tuned up (p. 36) by a fourth to γ, and from γ let it be tuned down by a fifth to δ.
τόνος ἄρα ὁ βδ.
Therefore, βδ is a whole tone.
πάλιν δὲ ἀπὸ τοῦ δ διὰ τεσσάρων ἐπιτετάσθω ἐπὶ τὸ ε, καὶ ἀπὸ τοῦ ε ἀνείσθω ἐπὶ τὸ διὰ πέντε.
And again, from δ let it be tuned up by a fourth to ε, and from ε let it be tuned down to [ζ] by a fifth.
τόνος ἄρα τὸ ζδ.
Therefore, ζδ is a whole tone.
δίτονος ἄρα τὸ ζβ. τόν τόν.
Therefore, ζβ is a ditone, tone, tone.
λιχανὸς ἄρα τὸ ζ.
Therefore, ζ is the lichanos.
ὁμοίως ἂν καὶ αἱ παρανῆται ληφθήσονται.
In like manner, the paranetes would also be obtained.