of the different modes, but as
they are mutually contradictory, one saying of a given mode that it is
bold and manly, while another calls it feeble and enervating, we may
leave this for the antiquarians to settle for themselves.

After Aristotle, there were several Greek theorists who devoted
themselves to mathematical computations, the favorite problem seeming
to be to find as many ways as possible of dividing the major fourth,
or the ratio 4:3, into what they called super-particular ratios--that
is to say, a series of fractions in which each numerator differed from
the denominator by unity. They had observed that all the ratios
discovered by Pythagoras had this character, 1/2, 2/3, 3/4, 8/9, and
they attributed magical properties to the fact, and sought to
demonstrate the entire theory of music by the production of similar
combinations. The latest writer of the Greek school was Claudius
Ptolemy, who lived at Alexandria about 150 A.D. In his work upon
harmony he gives a very large number of tables of fractions of this
kind--his own and those of all previous Greek theorists, and it is to
his book that we principally owe all the exact knowledge of Greek
musical theory which we possess. Among other computations, Ptolemy
gives the precise formula of the first four notes of the scale as we
now have it, but as this occurred only as one among many of a similar
character, and is in no way distinguished from any of the others by
any adjective implying greater confidence in it, we can only count it
as a lucky accident. The eminence that has been awarded to Ptolemy as
the original discoverer of the correct ratio of the major scale,
therefore, does not properly belong to him.

This will more clearly appear from the entire table of the various
determinations of the diatonic mode made by Ptolemy, taken from his
work. (Edition by John Wallis, Oxford, 1682, pp. 88 and 172.) He gives
no less than five of his own forms of diatonic genus, as follows: (The
fractions give vibration ratios.)

Soft diatonic, 8/7 x 10/9 x 21/20 = 4/3.
Medium diatonic, 9/8 x 8/7 x 28/27 = 4/3.
Intense diatonic, 10/9 x 9/8 x 16/15 = 4/3.
Equable diatonic, 10/9 x 11/10 x 12/11 = 4/3.
Diatonic diatonic, 9/8 x 9/8 x 256/243 = 4/3.

Among these there is no one that is correct or rational. The proper
ratios are given in the diatonic intense, but the large and small
steps stand in the wrong order. It is in Ptolemy's record of the
determ