Hoc in volumine haec opera continentur

The acute [part]. It is from the hypate meson lowest of the middle tetrachord to the paramese note next to the mese. The second [system] is comprised under the mesopycni notes of middle density, whose second tone is toward the acute part. It is from the parhypate meson note next to the lowest of the middle tetrachord to the third of the diezeugmenon disjunct tetrachord. The third is comprised under the oxypycnis notes of highest density, whose third tone is toward the acute part. It is indeed from the lichanos meson index finger note of the middle tetrachord to the paranete diezeugmenon note next to the lowest of the disjunct tetrachord.

The fourth is comprised under the barypycnis notes of lowest density, whose fourth tone is toward the grave part. It is indeed from the mese middle note to the nete diezeugmenon lowest note of the disjunct tetrachord or from the proslambanomenos added note to the hypate meson. In the diatonic genus, the first figure is the fourth, whose third is toward the grave part of that diapason octave; the species are seven. The first is comprised under the barypycnis, whose first tone is toward the acute part. It is from the hypate hypaton lowest note of the lowest tetrachord to the paramese; it is truly called the Mixolydian by the ancients. The second is comprised under the mesopycni, whose second tone is toward the acute part. It is from the parhypate hypaton note next to the lowest of the lowest tetrachord to the trite diezeugmenon third note of the disjunct tetrachord; it is called Lydian. The third is comprised under the oxypycnis, whose third tone is toward the acute part; it is from the lichanos hypaton index finger note of the lowest tetrachord to the paranete diezeugmenon; it is called Phrygian.

The fourth is comprised under the barypycnis, whose fourth tone is toward the acute part; it is from the hypate meson to the nete diezeugmenon; it is called Dorian. The fifth is comprised under the mesopycni, whose fifth tone is toward the acute part, is from the parhypate meson to the trite hyperbolaion third note of the highest tetrachord; it is named Hypolydian. The sixth is comprised under the oxypycnis, whose sixth tone is in the grave part, is from the lichanos meson to the paranete hyperbolaion note next to the lowest of the highest tetrachord; it is named Hypophrygian. The seventh is comprised under the barypycnis, whose first tone is in the grave part, is from the nete hyperbolaion lowest note of the highest tetrachord or from the proslambanomenos to the mese, called the Common and Locrian and Hypodorian. In the diatonic genus, the first species of diapason is that whose first [interval] is toward the grave and the fourth is toward the acute; it is a hemitonium semitone. The second, whose third is toward the grave, the first is toward the acute. The third, whose second is toward both. The fourth, whose first is toward the grave, the third is toward the acute. The fifth, whose fourth is toward the grave, the first is toward the acute. The sixth, whose third is toward the grave, the second is toward the acute. The seventh, whose fourth is toward the grave, the third is toward the acute. Again, these also [are referred] from the notes phthongis themselves to the same ones just as in harmony and the chromatic genus, and are called by the same names. But systems differ by the rational and irrational difference, for all that consist of rational intervals are distinguished from those that consist of irrational ones; for those that are rational are from rational [elements], but whatever are from irrational ones are irrational. But systems differ by the following and exceeding difference, which modulate the following notes by excess, and systems differ by synemmeni conjunct tetrachords and diezeugmeni disjunct tetrachords. Those that have the composition of tetrachords through the diesis quarter-tone have the diezeugmenon. It is indeed a synaphe conjunction, which is the contact of two tetrachords modulating consequentially, similar through the figure of the tone through the middle. There are three synaphae in all: middle, most acute, and most grave. The most grave is that which [arises] from that which