2. Intensity.
3. Quality (timbre).
By pitch is meant the highness or lowness of tone. It depends upon rate of vibration. If a body vibrates only 8 or 10 times per second no tone is heard at all: but if it vibrates regularly at the rate of 16 or 18 per second a tone of very low pitch is heard. If it vibrates at the rate of 24 the pitch is higher, at 30 higher still, at 200 yet higher, and when a rate of about 38,000 per second has been reached the pitch is so high that most ears cannot perceive it at all. The highest tone that can ordinarily be heard is the E♭ four octaves higher than the highest E♭ of the piano. The entire range of sound humanly audible is therefore about eleven octaves (rates 16-38,000), but only about eight of these octaves are utilized for musical purposes. The tones of the piano (with a range of 7-1/3 octaves) are produced by vibration rates approximately between 27 and 4224. In the orchestra the range is slightly more extended, the rates being from 33 to 4752.
Certain interesting facts regarding the relation between vibration-rates and pitches have been worked out: it has been discovered for instance that if the number of vibrations is doubled, the pitch of the resulting tone is an octave higher; i.e., if a string vibrating at the rate of 261 per second gives rise to the pitch c', then a string one-half as long and vibrating twice as rapidly (522) will give rise to the pitch c'', i.e., an octave higher than c'. In the same way it has been found that if the rate is multiplied by 5/4 the pitch of the tone will be a major third higher; if multiplied by 3/2, a perfect fifth higher, etc. These laws are often stated thus: the ratio of the octave to the fundamental is as two is to one; that of the major third as five is to four; that of the perfect fifth as three is to two, and so on through the entire series of pitches embraced within the octave, the ratio being of course the same for all octaves.
[9.] The intensity (loudness or softness) of tones depends upon the amplitude (width) of the vibrations, a louder tone being the result of vibrations of greater amplitude, and vice versa. This may be verified by plucking a long string (on cello or double-bass) and noting that when plucked gently vibrations of small amplitude are set up, while a vigorous pluck results in much wider vibrations, and, consequently, in a louder tone. It should be noted that the pitch of the tone is not affected by the change in amplitude of vibration.
The intensity of tones varies with the medium conveying them, being usually louder at night because the air is then more elastic. Tone intensity is also affected by sympathetic vibrations set up in other bodies. If two strings of the same length are stretched side by side and one set in vibration so as to produce tone the other will soon begin to vibrate also and the combined tone will be louder than if only one string produced it. This phenomenon is the basis of what is known as resonance (cf. body of violin, resonance cavities of nose and mouth, sounding board of piano, etc.).
[10.] Quality depends upon the shape (or form) of the vibrations which give rise to the tone. A series of simple vibrations will cause a simple (or colorless) tone, while complex vibrations (giving rise to overtones of various kinds and in a variety of proportions) cause more individualistic peculiarities of quality. Quality is affected also by the shape and size of the resonance body. (Cf. last part of [sec. 9] above.)
11. Practically every musical tone really consists of a combination of several tones sounding simultaneously, the combined effect upon the ear giving the impression of a single tone. The most important tone of the series is the fundamental, which dominates the combination and gives the pitch, but this fundamental is practically always combined with a greater or less number of faint and elusive attending tones called overtones or harmonics. The first of these overtones is the octave above the fundamental; the second is the fifth above this octave; the third, two octaves above the fundamental, and so on through the series as shown in the figure below. The presence of these overtones is accounted for by the fact that the string (or other vibrating body) does not merely vibrate in its entirety but has in addition to the principal oscillation a number of sectional movements also. Thus it is easily proved that a string vibrates in halves, thirds, etc., in addition to the principal vibration of the entire string, and it is the vibration of these halves, thirds, etc., which gives rise to the harmonics, or upper partials as they are often called. The figure shows Great C and its first eight overtones. A similar series might be worked out from any other fundamental.
