in the first movement of the _Scotch Symphony_,
where the interminable length of the portion in A minor (of all keys!)
is simply deadening in its effect. Compare also the _Prelude to the
Rheingold_; where, however--for dramatic purposes--to depict the world
as "without form and void" Wagner remains in the key of E-flat major
for some 150 measures!]

[Footnote 48: It is left to the teacher to explain, by the ratios
found in the overtones of the Harmonic Series, the validity of this
statement.]

[Footnote 49: Some modern theorists, _e.g._, Calvacoressi (see the New
Music Review for September, 1909) have thought that the dominant
relationship was "overworked." It is true that the great charm of
modern music is its freedom and boldness in modulation; but the
dominant keys can never be entirely abandoned, for the relationship
between them and a tonic is as elemental as that between the colors of
the spectroscope.]

Beginning with Beethoven, a modulation into what are known as the
_mediant_ keys became frequent; and is, in fact, a favorite change in
all modern music--the mediant keys being those situated half-way
between a Tonic and Dominant or a Tonic and Subdominant, _e.g._

[Music: Sub-mediant Mediant]

Anyone at all familiar with Beethoven's style will remember how often
his second theme, instead of following the more conventional line of
dominant relationship, is in a mediant key. Good examples may be found
in the first movement of the _Waldstein Sonata_ and in the first and
last movements of the 8th Symphony. A little thought will make clear
that the relationships just set forth include nearly all the possible
ones save those of 2nds and 7ths. Even into these apparently distant
keys, _e.g._, to D-flat major or to B major from C major, modulations
may easily be made by means of the "enharmonic"[50] relationship found
in that frequently used modern chord--the Augmented Sixth, _e.g._

[Music: C major B major C major D-flat major]

[Footnote 50: Two tones are said to be "enharmonic" when, although
written differently, they sound the same on an instrument of fixed
temperament like the pianoforte, or organ, _e.g._, D-sharp and E-flat,
E and F-flat. A violin, however, can make a distinction between such
notes and often does.]

Next to rhythm, modulation is the most stimulating and enchanting
element in music. No composition of any scope can be considered truly
great unless it abounds in beautiful modulations. Certain