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:
The figures below the lines denote how many times greater the number of vibrations is than that of the first tone. In the first octave we find only one tone; in the second, two; in the third, all the tones of the major chord with the minor seventh. In the fourth octave we find sixteen tones (which, however, we divide in our system of music into twelve). Likewise, we find in the fifth octave thirty-two tones, which number is doubled in the sixth. Hence, the Greeks had quarter and eighth tones, which we in our equal-tempered tuning have done away with.[ 9 ]
The production of a higher pitch in a tone rests in all sounding bodies upon the uniform law which we may observe in the strings of musical instruments, whose tones ascend either by greater tension, by shortening, or through a diminution of the density of the strings.
THE TIMBRE (KLANGFARBE) OF TONES
Strength and pitch were the first two distinctions of different tones. The third is the timbre. When we hear one and the same tone sounded successively upon a violin, trumpet, clarionet, oboe, upon a piano, or by a human voice, &c., although it is of the same strength and of the same pitch, yet the tone of all these instruments is different, and we very easily distinguish the instrument from which it comes. The changes of the timbre seem to be infinitely manifold; for, not to mention the fact that we have a multitude of different musical instruments, all which can give the same tone, letting alone also that different instruments of the same kind as well as different voices show certain differences of timbre, the very same tone can be given upon one and the same instrument, or by one and the same voice, with manifold differences of timbre.[ 10 ]
As now the strength of the tone is determined by the breadth of the vibrations, and the pitch by their number, so the varieties of timbre are ascribed to the different forms of the waves of vibration. For as the surface of the water is stirred differently by the falling into it of a stone, by the blowing over it of the wind, or the passing through it of a ship, &c., so the movements of the air take different shapes from sounding bodies. The movement proceeding from the string of a violin over which the bow is drawn, is different from those movements caused by the hammer of a piano or by a clarionet.
OVER-TONES (OBERTÖNE)
That timbre is dependent on the form of the vibrations is confirmed by Helmholtz, and acknowledged as so far correct that every different timbre requires a different vibratory form, but different forms sometimes correspond to nearly the same timbre. But how far the different forms of vibration correspond with different timbres, Helmholtz shows by a fact which has hitherto escaped the notice of physicists, although it forms the foundation of all music. We have learned by the stereoscope that we have two different views of every object, and compose a third view from those two. Just so the ear perceives different musical tones which come to our consciousness only as one tone.