Page:Popular Science Monthly Volume 16.djvu/539

Rh In this case the melody leaves the scale, but returns to it again, as shown by the notes marked which are raised. This method of altering the melody to obtain correct harmony is almost impossible to performers. It being understood that a note repeated or sustained is to be repeated or sustained at the same pitch, that it may become a pivot (ligature) for the harmonies to turn on, and form a standard of measurement. The errors would therefore more often be as follows:

5. When the chords are true, the melody is permanently out of proportion.

Here the ratios of each chord are prefixed to the letters, representing the musical notes, that the harmonies may be readily verified. At the fifth chord the key-note is seen at once to be changed, and the melody therefore to be untrue. Viewed vertically, all is correct; viewed horizontally, errors appear in all four lines. Such music can not be made correct from both points of view.

No idea is more firmly rooted in the minds of musicians than that of a fixed key-note. Whenever the pitch is changed the belief is universal that the chords have been out of tune. Even Helmholtz and other scientists are unaware of the fact that perfect harmony requires a moving key-note. It will probably surprise them as much as it would have surprised Ptolemy Philadelphus to learn that the sun is moving in the direction of the constellation Hercules.

For the solar system to be, as it were, in tune, the sun must move; for the harmonic system to be in tune, the key-note must move. In the last illustration the pitch of the key-note (C) was depressed in the ratio of 129 $3⁄5$ : 128. There would be three such depressions made in the first half of the melody, and by the same chords. There is no method by which the sum of the errors made in this direction may be atoned for by errors in the opposite direction. If, on repeating this half, the composer were to adopt the following harmonies, the key-note (C) would rise in the ratio of 63 : 64.