Second, Helmholtz discovered that the overtones always stand in the simplest mathematical relation to the fundamental tone,—in fact, are simple multiples of that tone, being two, three, four, and so on, times the number of vibrations of it.

This will be readily understood by considering the position of such related sounds when they are written upon the staff.

If we start with C in the bass as indicated in the staff, calling that the fundamental, then the notes that will represent the above ratios are those indicated by smaller notes, which are the overtones up to the ninth. The first overtone, being produced by twice the number of vibrations, must be the octave; the second, the fifth of the second octave; the third will be two octaves from the first, and so on: the number of vibrations of each of these notes being the number of the fundamental multiplied by its order in the series.

Taking C with 128 vibrations, we have for this series:—

128 ×1 = 128 = C fundamental.
128 ×2 = 256 = C´.
128 ×3 = 384 = G´.
128 ×4 = 512 = C´´.
128 ×5 = 640 = E´´.
128 ×6 = 768 = G´´.
128 ×7 = 896 = B´´♭.
128 ×8 = 1,024 = C´´´.
128 ×9 = 1,152 = D´´´.
128 ×10 = 1,280 = E´´´.

This series is continued up to the limits of hearing. Now, it appears that all instruments do not give the complete series: indeed, it is not possible to obtain them all upon some instruments. Each of them, however, when present helps in the general effect which we call quality. Sometimes the overtones are more prominent than the fundamental, as when a piano-wire is struck with a nail. It has always been noticed that it does not give out the sound that is wanted when it is struck in this way. Hence it is the art of an instrument-maker to so construct the instrument as to develop and re-enforce such tones as are pleasing, and to suppress the interfering and disagreeable overtones. Piano-makers learned by trial where was the proper place to strike the stretched wire in order to develop the most musical sound upon it; but no reason could be given until it was observed that striking it at a point about one-seventh or one-ninth its length from either end prevented the development of the objectionable overtones, the seventh and the ninth. Hence they can scarcely be heard in a properly constructed instrument. These overtones are very discordant with the lower sounds.

Organ-pipes have their specific qualities given to them by making them wide-mouthed, narrow-mouthed, conical, and so on; shapes which experience has determined give pleasing sounds with different qualities.

The violin is an instrument that seems to puzzle makers more than almost any other. Some of the old violins made two hundred years ago by the Amati family at Cremona are worth many times their weight in gold. Recent makers have tried in vain to equal them; but, when their ingenuity and skill have failed, they declare that age has much to do with such instruments, that age mellows the sounding quality of the violin. But the Cremona violins were just as extraordinary instruments when they left the hands of the makers as they are now; and the fame of the Amati family as violin-makers was over all Europe while they were living.

A good violin when well played gives an exquisite musical effect, and on account of its range and quality of tones it is the leading orchestral instrument, always pleasing and satisfying; but in unskilled hands even the best Cremona will give forth sounds that make one grieve that it was ever invented. Overtones of all sorts and with all degrees of prominence may be easily developed upon it: therefore the skilful player draws the bow at such a place upon the strings as to develop the overtones he wants, and suppress the ones not wanted. The usual rule is to draw the bow about an inch below the bridge; but the place for the bow depends upon where the fingers are that stop the strings, and also the pressure upon it. It requires an almost incredible amount of practice to be able to play a violin very well.