suspended at two-thirds of the one-third, i.e, one-ninth of the whole length above the center of oscillation, one-ninth above balances two-ninths below; the oscillating part is thus as it were, one-ninth shorter than at the center of oscillation, and gives rise to the center of velocity.
When a uniform dressed lath is held at the center of gravity and struck, it vibrates freely with a low tremor; when held at the center of velocity and struck at the center of gravity, it vibrates freely, and goes into large sections. But if it be struck at two-thirds of one-third from the other end, its own center of percussion, it has not the least tendency to vibrate. But if it be held at the center of velocity, and at its own center of percussion at the other end, and struck at the center of gravity, these two places will become points of rest, and the lath will move freely in all the other parts.
In oscillatory motion these centers act conjointly, but not simultaneously; but in vibratory motion these centers act simultaneously, and on each side of the center of gravity. [Scientific Basis and Build of Music, page 93]
In passing from one key to another in the fellowship of keys in a composition, the new key grows out of the top of the dominant and converts the old dominant into a tonic. The dominant and subdominant being at the opposite extremes of the key, with the tonic between them, are not related by affinity. This want of affinity makes an opening in the system for the new chord to come in by, and it, being related by affinity to the chord of the old dominant, which is now the new Tonic, comes in and establishes itself and the new key for the time. It is this gap between subdominant and dominant, along with the affinity existing between the new key and the old dominant, which makes this musical event to be so gracefully accomplished. This is what is called natural modulation, the passing for a time into another key in the course of a composition; and its abundant and habitual use in music, even in the simplest chorales, shows how natural and acceptable it is. The young student will find illustrations in the second lines of the Psalm tunes - Watchman, Sicily, Tranquility, Eaton, Birmingham, Jackson, Bethel, Bedford, and Sheffield. Take Watchman, for example, and let the young student follow carefully, noting each chord of the little passage, which we shall analyse for his help. It is by such practice that he will become by-and-by familiar with the kinship of keys and the legitimate resources of harmony.