e leader, and can proceed on the same plan
only by itself becoming the leader of a canon in augmentation.
Beethoven, in the fugues in his sonatas _op._ 106 and 110, adapted
augmentation and diminution to modern varieties of thematic expression,
by employing them in triple time, so that, by _doubling_ the length of
the original notes across this triple rhythm, they produce an entirely
new rhythmic expression. This does not seem to have been applied by any
earlier composer with the same consistency or intention.

The device of _inversion_ consists in the imitating part reversing every
interval of the leader, ascending where the leader descends and vice
versa. Its expressive power depends upon such subtle matters of the
harmonic expression of melody that its artistic use is one of the surest
signs of the difference between classical and merely academic music.
There are many melodies of which the inversion is as natural as the
original form, and does not strikingly alter its character. Such are,
for instance, the theme of Bach's _Kunst der Fuge_, most of Purcell's
contrapuntal themes, the theme in the fugue of Beethoven's sonata, _op._
110, and the eighth of Brahms's variations on a theme by Haydn. In such
cases inversion sometimes produces harmonic variety as well as a sense
of melodic identity in difference. But where a melody has marked
features of rise and fall, such as long scale passages or bold skips,
the inversion, if productive of good harmonic structure and expression,
may be a powerful method of transformation. This is admirably shown in
the twelfth of Bach's _Goldberg Variations_, in the fifteenth fugue of
the first book of his _Forty-eight Preludes and Fugues_, in the finale
of Beethoven's sonata, _op._ 106, and in the second subjects of the
first and last movements of Brahms's clarinet trio.

The only remaining canonic device which figures in classical music is
that known as _cancrizans_, in which the imitating part reproduces the
leader backwards. It is of extreme rarity in serious music; and, though
it sometimes happens by accident that a melody or figure of uniform
rhythm will produce something equally natural when read backwards, there
is only one example of its use that appeals to the ear as well as the
eye. This is to be found in the finale of Beethoven's sonata, _op._ 106,
where it is applied to a theme with such sharply contrasted rhythmic and
melodic features that with long familiarity a listener woul