aff, 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 x 1 = 128 = C fundamental.
128 x 2 = 256 = C'.
128 x 3 = 384 = G'.
128 x 4 = 512 = C''.
128 x 5 = 640 = E''.
128 x 6 = 768 = G''.
128 x 7 = 896 = B''[flat].
128 x 8 = 1,024 = C'''.
128 x 9 = 1,152 = D'''.
128 x 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