# Ptolemy Harmonics 2.2

Περὶ χρήσεως τοῦ κανόνος παρὰ τὸ καλούμενον ὄργανον ἑλικῶνα.

Αἱ μὲν οὖν περὶ τὰ γένη τῶν τετραχόρδων διαφοραὶ κατὰ τούτους ἡμῖν συνεστάθησαν τοὺς τρόπους διὰ τῆς τῶν ἀνισοτόνων φθόγγων ἀνακρίσεως καὶ παραβολῆς. γίνοιτο δ’ ἂν ἡ κατὰ τὸν ὀκτάχορδον κανόνα τοῦ διὰ πασῶν χρῆσις καὶ καθ’ ἕτερον τρόπον παρὰ τὸ καλούμενον ὄργανον ἑλικῶνα, πεποιημένον τοῖς ἀπὸ τῶν μαθημάτων εἰς τὴν ἔνδειξιν τῶν ἐν ταῖς συμφωνίαις λόγων οὕτωσί πως. ἐκτίθενται τετράγωνον ὡς τὸ ΑΒΓΔ καὶ διελόντες δίχα τὰς ΑΒ καὶ ΒΔ κατὰ τὰ Ε καὶ Ζ ἐπιζευγνύουσι μὲν τὰς ΑΖ καὶ ΒΗΓ, διάγουσι δὲ παρὰ τὴν ΑΓ διὰ μὲν τοῦ Ε τὴν ΕΘΚ, διὰ δὲ τοῦ Η τὴν ΛΗΜ. αὐτόθεν μὲν οὖν ἡ ΑΓ ἑκατέρας τῶν ΒΖ καὶ ΖΔ ὑπόκειται διπλασία καὶ ἔτι τούτων ἑκατέρα τῆς ΕΘ, ἐπεὶ καὶ ἡ ΑΒ τῆς ΑΕ, ὥστε καὶ τὴν ΑΓ τῆς μὲν ΕΘ τετραπλασίαν εἶναι, λοιπῆς δὲ τῆς ΘΚ ἐπίτριτον. δείκνυται δὲ ὅτι καὶ ἡ ΜΗ τῆς ΗΛ διπλασία ἐστίν, ἐπειδήπερ, ὡς μὲν ἡ ΔΓ πρὸς τὴν ΓΜ, οὕτως ἡ ΔΒ πρὸς τὴν ΗΜ, ὡς δὲ ἡ ΒΑ πρὸς τὴν ΑΛ, οὕτως ἡ ΒΖ πρὸς τὴν ΛΗ: καὶ διὰ τοῦτο ὡς ἡ ΒΔ πρὸς τὴν ΗΜ, οὕτως ἡ ΒΖ πρὸς τὴν ΛΗ, καὶ ἐναλλάξ, ὡς ἡ ΒΔ πρὸς τὴν ΒΖ, οὕτως ἡ ΜΗ πρὸς τὴν ΛΗ. γίνεται ἄρα ἡ ΑΓ καὶ τῆς μὲν ΗΜ ἡμιολία, τῆς δὲ ΗΛ τριπλασία, ὥστε διαταθεισῶν χορδῶν τεσσάρων ἰσοτόνων κατὰ τὰς αὐτὰς θέσεις τὰς τῶν ΑΓ καὶ ΕΚ καὶ ΛΜ καὶ ΒΔ εὐθειῶν, καὶ ὑπαχθέντος αὐταῖς κανονίου κατὰ τὴν τῆς ΑΘΗΖ θέσιν, ἐφαρμοσθέντων τε ἀριθμῶν τῇ μὲν ΑΓ τοῦ τῶν ιβʹ, τῇ δὲ ΘΚ τοῦ τῶν θʹ, τῇ δὲ ΗΜ τοῦ τῶν ηʹ, ἑκατέρᾳ δὲ τῶν ΒΖ καὶ ΖΔ τοῦ τῶν Ϛʹ, καὶ πάλιν τῇ μὲν ΛΗ τοῦ τῶν δʹ, τῇ δὲ ΕΘ τοῦ τῶν γʹ, ἀποτελεῖσθαι πάσας τὰς συμφωνίας καὶ τὸν τόνον, τῆς μὲν διὰ τεσσάρων καὶ κατὰ τὸν ἐπὶ γʹ λόγον συνισταμένης ὑπό τε τῶν ΑΓ καὶ ΘΚ καὶ ὑπὸ τῶν ΗΜ καὶ ΖΔ καὶ ὑπὸ τῶν ΛΗ καὶ ΘΕ, τῆς δὲ διὰ πέντε καὶ ἐν τῷ ἡμιολίῳ λόγῳ ὑπό τε τῶν ΑΓ καὶ ΗΜ καὶ ὑπὸ τῶν ΘΚ καὶ ΖΔ καὶ ὑπὸ τῶν ΒΖ καὶ ΛΗ, τῆς δὲ διὰ πασῶν καὶ κατὰ τὸν διπλάσιον λόγον ὑπό τε τῶν ΑΓ καὶ ΖΔ καὶ ὑπὸ τῶν ΗΜ καὶ ΛΗ καὶ ὑπὸ τῶν ΒΖ καὶ ΘΕ, τῆς δὲ διὰ πασῶν καὶ διὰ τεσσάρων ἐν τῷ τῶν ηʹ πρὸς τὰ γʹ λόγῳ ὑπὸ τῶν ΗΜ καὶ ΘΕ, τῆς δὲ διὰ πασῶν καὶ διὰ πέντε καὶ κατὰ τὸν τριπλάσιον λόγον ὑπὸ τῶν ΑΓ καὶ ΛΗ, τῆς δὲ δὶς διὰ πασῶν καὶ κατὰ τὸν τετραπλάσιον λόγον ὑπὸ τῶν ΑΓ καὶ ΕΘ, καὶ ἔτι τοῦ τόνου κατὰ τὸν ἐπὶ ηʹ λόγον ὑπὸ τῶν ΘΚ καὶ ΗΜ. Παρὰ δὴ τοῦτο τὸ ὄργανον ἐὰν ἐκθώμεθα παραλληλόγραμμον ἁπλῶς ὡς τὸ ΑΒΓΔ καὶ νοήσωμεν τὰς μὲν ΑΒ καὶ ΓΔ κατὰ τὰ ἀποψάλματα τῶν χορδῶν, τὰς δὲ ΑΓ καὶ ΒΔ κατὰ τοὺς ἄκρους φθόγγους τοῦ διὰ πασῶν, ἔπειτα προσεκβαλόντες τῇ ΓΔ ἴσην τὴν ΔΕ κατατέμωμεν ἀντὶ τῶν κανονίων τὴν ΓΔ πλευρὰν τοῖς οἰκείοις τῶν γενῶν λόγοις, ἐπὶ τοῦ Ε τὸ ὀξὺ πέρας ὑποτιθέμενοι, καὶ διὰ τῶν γινομένων ἐπ’ αὐτῆς τομῶν τείνωμεν τὰς χορδὰς παραλλήλους τε τῇ ΑΓ καὶ ἰσοτόνους ἀλλήλαις, καὶ τούτου γενομένου τὸν κοινὸν ἐσόμενον ὑπαγωγέα τῶν χορδῶν ὑποβάλωμεν αὐταῖς κατὰ τὴν ὑποζευγνῦσαν τὰ ΑΕ σημεῖα θέσιν ὡς τὸν ΑΖΕ, ποιήσομεν πάντα μήκη τῶν χορδῶν ἐν τοῖς αὐτοῖς λόγοις, ὥστε ἐπιδέχεσθαι τὴν τῶν ἐφηρμοσμένων τοῖς γένεσι λόγων ἀνάκρισιν. ἐπειδήπερ, ὡς αἱ ἀπὸ τοῦ Ε λαμβανόμεναι κατὰ τὴν ΓΔ πρὸς ἀλλήλας ἔχουσιν, οὕτω καὶ αἱ διὰ τῶν περάτων αὐτῶν ἀναγόμεναι παρὰ τὴν ΑΓ μέχρι τῆς ΑΖ ἕξουσι πρὸς ἀλλήλας, οἷον ὡς ἡ ΕΓ πρὸς τὴν ΕΔ, οὕτως ἡ ΓΑ πρὸς τὴν ΔΖ: διόπερ αὗται μὲν ποιήσουσι τὸ διὰ πασῶν, ὅτι διπλάσιος αὐτῶν ὁ λόγος. Ἐὰν δὲ ἀπολαβόντες πάλιν ἀπὸ τῆς ΓΔ τὴν μὲν ΓΗ κατὰ τὸ τέταρτον μέρος τῆς ΕΓ, τὴν δὲ ΓΘ κατὰ τὸ τρίτον τῆς αὐτῆς, ἀναστήσωμεν καὶ διὰ τῶν Η καὶ Θ χορδάς, ὡς τὰς ΗΚΛ καὶ ΘΜΝ ταῖς πρώταις ἰσοτόνους, ὥστε καὶ τήν τε ΑΓ τῆς μὲν ΗΚ γίνεσθαι ἐπίτριτον, τῆς δὲ ΘΜ ἡμιολίαν καὶ πάλιν τῆς ΔΖ τὴν μὲν ΘΜ ἐπίτριτον, τὴν δὲ ΗΚ ἡμιολίαν καὶ ἔτι τὴν ΗΚ τῆς ΘΜ ἐπὶ ηʹ, ποιοῦσι καὶ αὗται πρὸς ἀλλήλας τὰς ἀκολούθους τοῖς λόγοις συμφωνίας, τοῦ παραπλησίου παρακολουθήσοντος καὶ ἐπὶ τῶν μεταξὺ τοῖν τετραχόρδοιν λαμβανομένων τμημάτων ἐν τοῖς οἰκείοις τῶν ἀνακρινομένων λόγοις. ἔχει δ’ ὁ μὲν πρῶτος τρόπος παρὰ τοῦτον προχειρότερον τὸ μὴ δεῖν κινεῖν τὰς ἀπ’ ἀλλήλων διαστάσεις τῶν χορδῶν, οὗτος δὲ παρ’ ἐκεῖνον τό τε κοινὸν ἔχειν ὑπαγωγέα καὶ ἕνα καὶ κατὰ μίαν θέσιν καὶ ἔτι τὸ δύνασθαι καταβιβαζόμενον αὐτὸν διὰ τοῦ Ε, ὡς ἐπὶ τὴν ΞΟΕ θέσιν ὀξύτερον ποιεῖν ὅλον τὸν τόνον, μενούσης τῆς κατὰ γένος ἰδιότητος. ἐπεὶ καὶ ὡς ἡ ΓΑ φέρε εἰπεῖν πρὸς τὴν ΖΟΔ, οὕτως ἐστὶν ἡ ΞΓ πρὸς τὴν ΟΔ, καὶ ἐπὶ τῶν ἄλλων ὁμοίως. πάλιν τ’ αὖ κατασκελέστερον ὁ μὲν πρότερος ἔχει τρόπος παρὰ τοῦτον τὸ πλέονα δεῖν κινεῖν ὑπαγωγίδια καθ’ ἑκάστην ἁρμογήν, οὗτος δὲ παρ’ ἐκεῖνον τὸ τὰς χορδὰς ὅλως παραφέρειν, καὶ μηκέτι κατ’ ἴσας αὐτῶν διαστάσεις, ἀλλὰ πολλαχῇ μακρῷ διαφερούσας συντελεῖσθαι τὰς τῶν ἐπιψαύσεων μεταβάσεις.

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