f 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
gi