. But from the fifth to the octave gives the interval of the fourth, therefore he permitted this combination also.
From the works of Bœthius (circa 400) and others, he had derived and accepted the Pythagorean division of the scale, making thirds and sixths dissonant intervals; and so his perfect chord (from which our later triad gets its name of perfect) was composed of a root, fifth or fourth, and octave.
Hucbald, as I have already explained, changed the Greek tone system somewhat by arranging it in four regular disjunct tetrachords, namely:
This system permitted the addition of a fifth to each note indiscriminately, and the fifths would always be perfect; but in regard to the octaves it was faulty, for obvious reasons. As his system of notation consisted of merely writing T for tone and S for semitone between the lines of his staff, it was only necessary to change the order of these letters for the octave at the beginning of each line. With the fourth, however, this device was impossible, and therefore he laid down the rule that when the voices proceeded in fourths, and a discord (or augmented fourth) was unavoidable, the lower voice was to remain on the same note until it could jump to another fourth forming a perfect interval:
This at least brought into the harmony an occasional third, which gradually became a recognized factor in music.
We probably know that the year 1000 was generally accepted as the time when the world was to come to an end. In the Bibliothèque Nationale in Paris there is a manuscript containing the prophecy which had been handed down for many centuries; also the signs for the notes to which it was to be sung, viz.:
[[MIDI]]
