tedious, is not difficult if followed progressively. First, we will
consider tone relationship in connection with relative string length.
Students who have small stringed instruments, guitar, violin, or
mandolin, may find pleasure in demonstrating some of the following
facts thereupon.

One-half of any string will produce a tone exactly an octave above
that yielded by its entire length. Harmonic tones on the violin are
made by touching the string lightly with the finger at such points as
will cause the string to vibrate in segments; thus if touched exactly
in the middle it will produce a harmonic tone an octave above that of
the whole string.

Two-thirds of the length of a string when stopped produces a tone a
fifth higher than that of the entire string; one-third of the length
of a string on the violin, either from the nut or from the bridge, if
touched lightly with the finger at that point, produces a harmonic
tone an octave higher than the fifth to the open tone of that string,
because you divide the string into three vibrating segments, each of
which is one-third its entire length. Reason it thus: If two-thirds of
a string produce a fifth, one-third, being just half of two-thirds,
will produce a tone an octave higher than two-thirds. For
illustration, if the string be tuned to 1C, the harmonic tone produced
as above will be 2G. We might go on for pages concerning harmonics,
but for our present use it is only necessary to show the general
principles. For our needs we will discuss the relative length of
string necessary to produce the various tones of the diatonic scale,
showing ratios of the intervals in the same.

In the following table, 1 represents the entire length of a string