the last A[b] but one, in the standard scale of seven and a third
octaves of which the last key is C. Sharpen but slightly, and increase
the degree of sharpening but little as you proceed.

~Tuning the Bass.~--In tuning the bass, listen for the beats only, in
bringing up the octaves. It is sometimes well to try the string tuned,
with its fifth, but the octave in the bass should suffice, as the
vibrations are so much slower here that if you listen acutely the
octave beats will guide you.

It is not necessary to pull the strings higher than the pitch at which
they are to stand. Learn to pull them up gradually and in a way that
will "render" the string over the bridges, which is an easy thing to
do, the strings being so much heavier here than elsewhere. Never leave
a bass string the slightest amount too sharp. As flatness is so
obnoxious in the treble, just so is sharpness in the bass, so if there
must be any variation in any bass tone let it be flat; but aim at
perfect octaves throughout the bass.

~False Waves.~--We say "false waves" for want of a better name. You
will find a string occasionally that will give forth waves or beats so
similar to the real ones that it takes a practiced ear to distinguish
the difference. Where a unison contains a string of this kind, select
some other string by which to tune the interval, and leave the bad
string until the last; you may then find difficulty in being able to
tell when you have it in unison. The cause may be a twisted string, a
fault in the string by imperfect drawing of the wire, or in the
construction of the sound-board.

In the low bass tones, a kind of false waves are always present, and
will annoy the tuner long after he has been in regular practice. They
are, however, of a different nature from the true waves in that they
are of a metallic timbre and of much greater rapidity than the latter.
Close attention will generally enable the tuner to distinguish between
them. They are caused by what is known as "harmonics" or "over-tones";
the string vibrating in fractional segments.

False waves will occur in an annoying degree when the tuner sets a
mute on a nodal point in the string; it will cause the muted string to
sound a real harmonic tone. This does not happen in the upright, as
the mutes are set so near the end of the string as to preclude this
possibility. In the square, however, it very frequently happens, as
there are so many nodes between the dampers and the b