mallest step on the piano, called the "semitone", is one-twelfth of
an octave; but it must not be supposed that this is the smallest
difference that can be perceived. A large proportion of people can
observe a difference of four vibrations, and keen ears a difference of
less than one vibration; whereas the semitone, at middle C, is a step
of about sixteen vibrations.

_Mixture of different wave-lengths_, which in light causes difference
of saturation, may be said in sound to cause difference of purity. A
"pure tone" is the sensation aroused by a stimulus consisting wholly
of waves of the same length. Such a stimulus is almost unobtainable,
because every sounding body gives off, along with its fundamental
waves, other waves shorter than the fundamental and arousing tone
sensations of higher pitch, called "overtones". A piano string which,
vibrating as a whole, gives 260 vibrations per second (middle C), also
vibrates at the same time in halves, thus giving 520 vibrations per
second; in thirds, giving 780 per second; and in other smaller
segments. The whole stimulus given off by middle C of the piano is
thus a compound of fundamental and overtones; and the sensation
aroused by this complex stimulus is not a "pure tone" but a blend of
fundamental tone and overtones. By careful attention and training, we
can "hear out" the separate overtones from the total blend; but
ordinarily we take the blend as a unit (just as we take the taste of
lemonade as a unit), and hear it simply as middle C of a particular
quality, namely the piano quality. Another instrument will give a
somewhat different combination of overtones in the stimulus, and that
means a different quality of tone in our sensation. We do not
ordinarily analyze these complex blends, but we distinguish one from
another perfectly well, and thus can tell whether a piano or a cornet
is playing. The difference between different instruments, which we
have spoken of as a {231} difference in quality or purity of tone, is
technically known as _timbre_; and the timbre of an instrument depends
on the admixture of shorter waves with the fundamental vibration which
gives the main pitch of a note.

Akin to the timbre of an instrument is the _vowel_ produced by the
human mouth in any particular position. Each vowel appears to consist,
physically, of certain high notes produced by the resonance of the
mouth cavity. In the position for "ah", the cavity gives a certain
tone; in the posi