ean circle [O.] or [O], so Hucbald,
in choosing his series of concords or sounds that harmonize
well together, took the first three notes of the overtones of
every sonorous fundamental, or, to express it differently, of
the series of natural harmonics, that is to say, he admitted
the octave and fifth: [F: g, d g]. But from the fifth to the
octave gives the interval of the fourth, therefore he permitted
this combination also.

From the works of Boethius (_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:

[F: (g, a, b-, c) (d e f g) G: (a b c' d') (e' f+' g' a')]

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:

[F: {g b} {g b} {g a} {g b} {d a} {d g} {c f} {c e} {a, d} {g, c}]

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
_Bibliotheque 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.:

[Figure 07]

The text is:

The Judge will speak and the earth shall tremble
with awe. The stars shall be destroyed and the glory
of the moon shall die, the mountains shall be crushed
and the world with all in it shall utterly perish.

With the opening of the eleventh century, such was the reli