of
mathematics, the simplicity of the elements involved; but the intricacy
of their details and the subtlety of their expression may easily pass
the limits of popularity, while art of a much more complex nature may
masquerade in popular guise; just as mathematical science is seldom
popularized, while biology masquerades in infant schools as "natural
history." Here, however, the resemblance between counterpoint and
mathematics ends, for the simplicity of genuine contrapuntal style is a
simplicity of emotion as well as of principle; and if the style has a
popular reputation of being severe and abstruse, this is largely because
the popular conception of emotion is conventional and dependent upon an
excessive amount of external nervous stimulus.

_1. Canonic Forms and Devices._

In the _canonic_ forms, the earliest known in music as an independent
art, the laws of texture also determine the shape of the whole, so that
it is impossible, except in the light of historical knowledge, to say
which is prior to the other. The principle of canon being that one voice
shall reproduce the material of another note for note, it follows that
in a composition where all parts are canonic and where the material of
the leading part consists of a pre-determined melody, such as a
Gregorian chant or a popular song there remains no room for further
consideration of the shape of the work. Hence, quite apart from their
expressive power and their value in teaching composers to attain
harmonic fluency under difficulties, the canonic forms played the
leading part in the music of the 15th and 16th centuries; nor indeed
have they since fallen into neglect without grave injury to the art. But
strict canon soon proved inadequate, and even dangerous, as the sole
regulating principle in music; and its rival and cognate principle, the
basing of polyphonic designs upon a given melody to which one part
(generally the tenor) was confined, proved scarcely less so. Nor were
these two principles, the canon and the _canto fermo_, likely, by
combination in their strictest forms, to produce better artistic results
than separately. Both were rigid and mechanical principles; and their
development into real artistic devices was due, not to a mere increase
in the facility of their use, but to the fact that, just as the
researches of alchemists led to the foundations of chemistry, so did the
early musical puzzles lead to the discovery of innumerable harmonic and
melo