CHAPTER 3

THE OCTOCHORD

Now Pythagoras, first of all, left the middle sound by conjunction, being itself compared to the two extremes, should render only a diatessaron harmony, both to the neate and to the hypate. But that we might have greater variety, the two extremes making the fullest concord each to other, that is to say the concord of diapason, which consists in a double proportion.597 Inasmuch as it could not be done by two Tetrachords, he added an eighth sound, inserting it between the mese and paramese—setting it from the mese a whole tone, and from the paramese a semitone. So that which was formerly the paramese in the Heptachord is still the third from the neate, both in name and place. But that which was now inserted is the fourth from the neate, and has a harmony unto it of diatessaron—which before, the mese had unto the hypate.⁵⁹⁸

But the tone between them, that is the mese, and the inserted called the paramese, instead of the former, to whichever Tetrachord it be added, whether to that which is at the hypate, being of the lower; or to that of the neate, being of the higher; will render diapente concord. This is either way a system consisting both of the Tetrachord itself, and the additional tone; as the diapente-proportion (viz. sesquialtera) is found to be a system of sesquitertia, and sesquioctava; the Tone therefore is sesquioctava.⁵⁹⁹ Thus the interval of four chords, and of five, and of both conjoined together, called diapason, and the tone inserted between the two Tetrachords, being after this manner apprehended by Pythagoras, were determined to have this proportion in numbers.