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 sounding the tone C. The other tones of the ascending major scale require strings of such fractional length as are indicated by the fractions beneath them.  By taking accurate measurements you can demonstrate these figures upon any small stringed instrument.

Funda- | Major | Major | Perfect | Perfect | Major |  Major  |  Oc- |
mental |Second | Third | Fourth  |  Fifth  | Sixth | Seventh | tave |
|       |       |         |         |       |         |      |
C   |   D   |   E   |    F    |    G    |   A   |    B    |  C   |
1   |  8/9  |  4/5  |   3/4   |   2/3   |  3/5  |   8/15  | 1/2  |

To illustrate this principle further and make it very clear, let us suppose that the entire length of the string sounding the fundamental C is 360 inches; then the segments of this string necessary to produce the other tones of the ascending major scale will be, in inches, as follows: 

C  |  D  |  E  |  F  |  G  |  A  |  B  |  C  |
360 | 320 | 288 | 270 | 240 | 216 | 192 | 180 |

Comparing now one with another (by means of the ratios expressed by their corresponding numbers) the intervals formed by the tones of the above scale, it will be found that they all preserve their original purity except the minor third, D-F, and the fifth, D-A.  The third, D-F, presents itself in the ratio of 320 to 270 instead of 324 to 270 (which latter is equivalent to the ratio of 6 to 5, the true ratio of the minor third).  The third, D-F, therefore, is to the true minor third as 320 to 324 (reduced to their lowest terms by dividing both numbers by 4, gives the ratio of 80 to 81).  Again, the fifth, A-F, presents itself in the ratio of 320 to 216, or (dividing each term by 4) 80 to 54; instead of 3 to 2 (=81 to 54—­multiplying each term by 27), which is the ratio of the true fifth.  Continuing the scale an octave higher, it will be found that the sixth, F-D, and the fourth, A-D, will labor under the same imperfections.