4Harmony is an obscure and difficult musical science, but most difficult to those who are not acquainted with the Greek language; because it is necessary to use many Greek words to which there are none corresponding in Latin. I will therefore explain, to the best of my ability, the doctrine of Aristoxenus, and annex his diagram, and will so designate the place of each tone, that a person who studiously applies himself to the subject may very readily understand it.
2The inflexion of the voice is two-fold; first, when it is monotonous, second, when it proceeds by intervals. The first is not limited by cadences at the close, nor in any other place; no perceptible difference of tone being discoverable between its beginning and its ending, the time between each sound is however distinctly marked, as in speaking, when we pronounce the words, sol, lux, flos, nox. Herein the ear does not perceive any difference of tone between the beginning and the ending, by the voice rising higher or descending lower; neither, that from a high pitch it becomes lower, nor the contrary. But when the voice moves by intervals, it is differently inflected, being sometimes at a high pitch, and sometimes at a low one, and resting at different times on different tones; by doing which with quickness and facility, it appears unfixed. Thus in singing, the variety of inflexion produces an air. In short, by the use of different intervals, the tones are so marked and determined, that we perceive the pitch at which it begins, and that at which it finishes, though the intermediate tones are not heard.
3There are three sorts of modulation, the enharmonic (ἁρμονία), the chromatic (χρῶμα), and the diatonic (διάτονος), so called by the Greeks. The enharmonic is so constructed by art, as to be full of majesty and pathos. The chromatic by the skilful contrivance and closeness of its intervals has more sweetness. The diatonic, whose intervals are more simple, is most natural. The disposition of the tetrachords, in these genera, are dissimilar. The enharmonic tetrachord consists of two dieses, and two whole tones; a diesis being the fourth part of a tone, and two of them consequently equal to a semitone. In the chromatic tetrachord, there are two consecutive semitones, and the third interval contains three semitones. The diatonic tetrachord has two consecutive tones, and an interval of a semitone. Thus in each genus, the whole tetrachord is equal to two whole tones and a semitone. But the intervals in each genus, differ when considered separately.
4For nature has made the divisions of tones, semitones, and tetrachords, and has established those proportions of the intervals, by which workmen are guided in making and assigning their just proportions to instruments.
5Each genus consists of eighteen sounds, which the Greeks call φθόγγοι (phthongi). Of these, eight sounds in each of the genera, vary neither in sound nor situation. The remaining ten in each are not common to the other two genera. Those which do not vary, contain between them the variable sounds, and are the limits of the tetrachords in all the genera. Their names are as follow: proslambanomenos, hypatè hypatôn, hypatè mesôn, mesè, netè synèmmenôn, paramesè, netè diezeugmenôn, netè hyperbolæôn. The variable, which lie between those that are not variable, change their places according to the genus. Their names are parhypatè hypatôn, lichanos hypatôn, parhypatè mesôn, lichanos mesôn,tritè synèmmenôn, paranetè synèmmenôn, tritè diezeugmenôn, paranetè diezeugmenôn, tritè hyperbolæôn, paranetè hyperbolæôn.
6Those sounds which shift their places, change also their nature, and are at different intervals, as, for instance, the interval between hypatè and parhypatè, which in the enharmonic genus is only a diesis or quarter tone, is in the chromatic genus a semitone. So the lichanos is only a semitone distant from the hypatè in the enharmonic genus; whereas in the chromatic it is two semitones distant, and in the diatonic three semitones. Thus, the ten sounds, by their situation in the different genera, make three different sorts of melody.
7There are five tetrachords. The Greeks call the lowest ὕπατον (hypaton); the second, which is in the middle, μέσον (meson). The third, which is joined to the two preceding, is called συνημμένον (synèmmenon). The fourth, which is disjoined, called διεζευγμένον (diezeugmenon). The fifth, which is the highest, the Greeks call ὑπερβόλαιον(hyperbolæon). The natural consonances, which the Greeks call συμφωνίαι (symphoniæ), are six in number; diatessarôn (fourth), diapente (fifth), diapasôn (octave), diapasôn with diatessarôn(eleventh), diapasôn with diapente (twelfth), and disdiapasôn (fifteenth).
8These names are given them from the number of tones which the voice passes through in going to them, counting that on which the voice begins as one; thus, moving through them to the fourth sound is called diatessarôn; to the fifth, diapente. to the eighth diapasôn, to the eleventh diapasôn with diatessarôn, to the twelfth diapasôn with diapente, to the fifteenth disdiapasôn.
9For between two intervals, either in a melody sung by a voice, or played on a stringed instrument, neither with the third, sixth nor seventh can there be consonances, but only, as above shewn, with the diatessarôn and diapente up to the diapasôn do natural consonances arise, and those are produced by an union of those sounds which the Greeks call φθόγγοι (phthongi).