§18Αἱ παρυπάται καὶ αἱ τρίται οὐ διαιροῦσι τὸ πυκνὸν εἰς ἶσα.
The parypatai and the tritai do not divide the pyknon into equal parts.
Ἔστω γὰρ μέση μὲν ὁ β, λιχανὸς ζ δ γ ε β δὲ ὁ γ, ὑπάτη δὲ ὁ δ.
For let the mese be β, the lichanos be γ, and the hypate be δ.
ἀνείσθω ἀπὸ τοῦ β διὰ πέντε ἐπὶ τὸ ζ.
Let it be tuned down from β by a fifth to ζ.
τόνος ἄρα ὁ ζδ.
Therefore, ζδ is a whole tone.
καὶ ἀπὸ τοῦ ζ διὰ τεσσάρων ἐπιτετάσθω ἐπὶ τὸ ε.
And from ζ, let it be tuned up by a fourth to ε.
τόνος ἐστὶν ἄρα τὸ ζδ διάστημα καὶ τὸ βε.
Therefore, the interval ζδ is a whole tone, and so is βε.
κοινὸν προσκείσθω τὸ δγ.
Let the common [interval] δγ be added.
τὸ ἄρα ζε ἶσόν ἐστι τῷ δβ.
Therefore, ζε is equal to δβ.
διὰ τεσσάρων δὲ τὸ ζε·
And ζε is a fourth.
οὐκ ἄρα μέσος ἀνάλογον ἐμπίπτει τις τῶν ζε· ἐπιμόριον γὰρ τὸ διάστημα.
Therefore, no mean proportional falls between [the terms of] ζε; for the interval is superparticular.
ξδ καὶ ἔστιν ἴσος ὁ δβ τῷ ζε· οὐκ ἄρα οβ τοῦ δ μέσος ἐμπεσεῖται, ὅ ἐστιν ἀπὸ ὑπάτης ἐπὶ λιχανόν.
And δβ is equal to ζε; therefore, no mean will fall into δβ, which is [the interval] from hypate to lichanos.
οὐκ ἄρα ἡ παρυπάτη διελεῖ τὸ πυκνὸν εἰς ἶσα.
Therefore, the parhypate will not divide the pyknon into equal parts.
ὁμοίως οὐδὲ ἡ τρίτη. (p. 37. ) §19Τὸν κανόνα καταγράψαι κατὰ τὸ καλούμενον ἀμετάβολον σύστημα.
Likewise, neither will the trite. (p. 37.) To graduate the monochord according to the so-called immutable system.
Ἔστω τοῦ κανόνος μῆκος, ὃ καὶ τῆς χορδῆς τὸ αβ, καὶ διῃρήσθω εἰς τέσσαρα ἶσα κατὰ τὰ γδε.
Let the length of the monochord, which is also that of the string, be αβ, and let it be divided into four equal parts at γ, δ, and ε.
ἔσται ἄρα ὁ βα βαρύτατος ὢν φθόγγος βόμβυξ.
Therefore, βα, being the lowest sound, will be the bombyx.
οὗτος δὲ ὁ αβ τοῦ γβ ἐπίτριτός ἐστιν, ὥστε ὁ γβ τῷ αβ συμφωνήσει διὰ τεσσάρων ἐπὶ τὴν ὀξύτητα.
And this αβ is sesquitertian of γβ, so that γβ will concord with αβ at the fourth towards the high pitch.
καὶ ἔστιν ὁ αβ προσλαμβανόμενος· ὁ ἄρα γβ ἔσται ὑπάτων διάτονος.
And αβ is the proslambanomenos; therefore, γβ will be the hypaton diatonos.
πάλιν ἐπεὶ ὁ αβ τοῦ βδ ἐστὶ διπλοῦς, συμφωνήσει τῇ διὰ πασῶν καὶ ἔσται ὁ βδ μέση.
Again, since αβ is double of βδ, it will concord at the octave, and βδ will be the mese.
πάλιν ἐπεὶ τετραπλάσιός ἐστιν ὁ αβ τοῦ εβ, ἔσται ὁ εβ νήτη ὑπερβολαίων.
Again, since αβ is quadruple of εβ, εβ will be the nete hyperbolaion.
ἔτεμον τὸν γβ δίχα κατὰ τὸ ζ.
I cut γβ in half at ζ.
καὶ ἔσται διπλάσιος ὁ γβ τοῦ ζβ, ὥστε συμφωνεῖν τὸν γβ πρὸς τὸν ζβ διὰ πασῶν· ὥστε εἶναι τὸν ζβ νήτην συνημμένων.
And γβ will be double of ζβ, so that γβ concords with ζβ at the octave; so that ζβ is the nete synemmenon.
ἀπέλαβον τοῦ δβ τρίτον μέρος τὸ δη.
I cut off a third part of δβ, which is δη.
καὶ ἔσται ἡμιόλιος ὁ δβ τοῦ ηβ, ὥστε συμφωνήσει ὁ δβ πρὸς τὸν ηβ ἐν τῷ διὰ πέντε· ὁ ἄρα ηβ νήτη ἔσται διεζευγμένων.
And δβ will be sesquialter of ηβ, so that δ β will concord with ηβ at the fifth; therefore, ηβ will be the nete diezeugmenon.
ἔθηκα τῷ ηβ ἶσον τὸν ηθ, ὥστε ὁ θβ πρὸς τὸν ηβ συμφωνήσει διὰ πασῶν, ὡς εἶναι τὸν θβ ὑπάτην μέσων.
I placed ηθ equal to ηβ, so that θβ will concord with ηβ at the octave, so that θβ is the hypate meson.
ἔλαβον τοῦ θβ τρίτον μέρος τὸ θκ.
I took a third part of θβ, which is θκ.
καὶ ἔσται ἡμιόλιος ὁ θβ τοῦ κβ, ὥστε εἶναι τὸν κβ παράμεσον. (p. 39. ) ἀπέλαβον τῷ κβ ἶσον τὸν καὶ γενήσεται ὁ λβ ὑπάτη βαρεῖα.
And θβ will be sesquialter of κβ, so that κβ is the paramese. (p. 39.) I cut off [κλ] equal to κβ, and λβ will be the low hypate.
ἐσονται ἄρα εἰλημμένοι ἐν τῷ κανόνι πάντες οἱ ἑστῶτες φθόγγοι τοῦ ἀμεταβόλου συστήματος. α λ γ δ κ ζ η ε προσλαμβ.
Therefore, all the standing sounds of the immutable system will have been obtained on the monochord. α, λ, γ, δ, κ, ζ, η, ε.
ὑπάτη ὑπ. λιχ. ὑπ διάτ. ὑπάτη μέσ. Μέση παραμέση νήτη συν. νήτη διεζ. νήτη ὑπβ.
Proslambanomenos, hypate hypaton, hypaton diatonos, hypate meson, mese, paramese, nete synemmenon, nete diezeugmenon, nete hyperbolaion.
§20Λοιπὸν δὴ τοὺς φερομένους λαβεῖν.
It remains, then, to obtain the movable sounds.
Ἔτεμον τὸν ε β εἰς ὀκτὼ καὶ ἑνὶ αὐτῶν ἶσον ἔθηκα τὸν εμ, ὥστε τὸν μβ τοῦ εβ γενέσθαι ἐπόγδοον.
I cut εβ into eight, and placed εμ equal to one of them, so that μβ becomes sesquioctave of εβ.
καὶ πάλιν διελὼν τὸν μβ εἰς ὀκτὼ ἑνὶ αὐτῶν ἶσον ἔθηκα τὸν νμ· τόνῳ ἄρα βαρύτερος ἔσται ὁ νβ τοῦ βμ, ὁ δὲ μβ τοῦ βε, ὥστε ἔσται μὲν ὁ νβ τρίτη ὑπερβολαίων, ὁ δὲ μβ ὑπερβολαίων διάτονος.
And again, dividing μβ into eight, I placed νμ equal to one of them; therefore, νβ will be lower by a whole tone than βμ, and μβ than βε, so that νβ will be the trite hyperbolaion, and μβ the hyperbolaion diatonos.
ἔλαβον τοῦ νβ τρίτον μέρος καὶ ἔθηκα τὸν νξ, ὥστε τὸν ξβ τοῦ νβ εἶναι ἐπίτριτον καὶ διὰ τεσσάρων συμφωνεῖν ἐπὶ τὴν βαρύτητα καὶ γενέσθαι τὸν ξβ τρίτην διεζευγμένων.
I took a third part of νβ and placed νξ, so that ξβ is sesquitertian of νβ and concords at the fourth towards the low pitch, and ξβ becomes the trite diezeugmenon.
πάλιν τοῦ ξβ λαβὼν ἥμισυ μέρος ἔθηκα τὸν ξο, ὥστε διὰ πέντε συμφωνεῖν τὸν οβ πρὸς τὸν ξβ· ὁ ἄρα οβ ἔσται παρυπάτη μέσων.
Again, taking a half part of ξβ, I placed ξο, so that οβ concords with ξβ at the fifth; therefore, οβ will be the parhypate meson.
καὶ τῷ ξο ἶσον ἔθηκα τὸν οπ, ὥστε γενέσθαι τὸν πβ παρυπάτην ὑπάτων.
And I placed οπ equal to ξο, so that πβ becomes the parhypate hypaton.
ἔλαβον δὴ τοῦ β τέταρτον μέρος τὸν γρ, ὥστε γενέσθαι τὸν ρβ μέσων διάτονον.
Then I took a fourth part of [γβ], namely γρ, so that ρβ becomes the meson diatonos.
προσλαμβανόμενος α
ὑπάτη ὑπάτων
παρυπ ὑπάτων
λιχανὸς ὑπάτων
ὑπάτη μέσων
παρυπάτη μέσων
λιχανὸς μέσων
Μέση
παραμέση
τρίτη διεζευγμένων
παρανήτη διεζευγμένων
νήτη διεζευγμένων
τρίτη ὑπερβολαίων
παρανήτη ὑπερβολαίων
νήτη ὑπερβολαίων
Proslambanomenos α hypate hypaton parhypate hypaton lichanos hypaton hypate meson parhypate meson lichanos meson mese paramese trite diezeugmenon paranete diezeugmenon nete diezeugmenon trite hyperbolaion paranete hyperbolaion nete hyperbolaion