The sounds produced by such periodic agitations of the air have
three peculiar properties: 1. STRENGTH, 2. PITCH, 3. TIMBRE.
The strength of the tone depends on the greater or less breadth
of its vibrations, that is, of the waves of sound, the higher or
lower pitch of the tones upon the number of the vibrations; that
is, the tones are always higher the greater the number of the
vibrations, or lower the less the number of the vibrations.
A second is used as the unit of time, and by number of vibrations
is understood the number of vibrations which the sounding body
gives forth in a second of time. The tones used in music lie
between 40 and 4000 vibrations per second, in the extent of seven
octaves. The tones which we can perceive lie between 16 and 38,000
vibrations to the second, within the compass of eleven octaves.
The later pianos usually go as low as C₁ with 33, or even to A₂
with 27½ vibrations; mostly as high as a⁴ or c⁵, with 3520 and
4224 vibrations. The one lined a¹, from which all instruments
are tuned, has now usually 440 to 450 vibrations to the second
in England and America. The French Academy, however, has recently
established for the same note 435 vibrations, and this lower
tuning has already been universally introduced in Germany.[8]
The high octave of a tone has in the same time exactly double
the number of vibrations of the tone itself. Suppose, therefore,
that a tone has 50 vibrations in a second, its octave has 100
in the same time; i.e., twice as many. The octave above this has
200 vibrations, &c. The Pythagoreans knew this acoustic law of
the ascending tones, and that the octave of a tone had twice as
many vibrations in a second as the tone itself, and that the
fifth above the first octave had three times as many; the second
octave, four times; the major third above the second octave,
five times as many; the fifth of the same octave, six times; the
small seventh of the same octave, seven times. In notation it
would be thus, if we take as the lowest note C, for example: