A C O U 
each end, like the bridge of a fiddle, and fcrewed up with 
fcrew-pins. The firings are all tuned to the fame note ; 
and the inftrument is placed in fome current of air, fuch 
as a window with the lafii juft raifed. The air brufhing 
the firings, will excite different tones of found, and fome- 
times bring out all the tones in full concert, fo as to occa- 
fion very pleafing gradations of found. 
It remains, lafily, to confider (by this theory) what the 
tnuficians call the ftrength and foftnefs of found. In vi¬ 
brating elaftic firings, the loudnefs of the tone is in pro¬ 
portion to the deepnefs of the note; that is, in two firings, 
all things in other circumfiances alike, the deepeft tone 
will be loudeft. In mufical inftruments upon a different 
principle, as in the violin, it is otherwife; the tones are 
made in fuch inftruments by a number of fmall vibra¬ 
tions crowded into one ftroke. The rofined bow, for in¬ 
fiance, being drawn along a firing, its roughnefs catches 
the firing at very fmall intervals, and excites its vibrations. 
In this inftrument, therefore, to excite loud tones, the 
bow mull be drawn quick, and this will produce the great- 
eft number of vibrations. But it mull be obferved, that 
the more quick the bow paffes over the firing, the lefs apt 
will the roughnefs of its furface be to touch the firing at 
every inftant: to remedy this, therefore, the bow mull be 
prefled the harder as it is drawn quicker, and thus its full- 
eft found will be brought from the inftrument. If the 
fwiftnefs of the vibrations in an inftrument thus rubbed 
upon, exceed the force of the deeper found in another, 
then the fwift vibrations will be heard at a greater dis¬ 
tance, and as much farther off as the fwiftnefs in them 
exceeds the force in the other. 
By the fame theory (it is alleged) may all the phenome¬ 
na of mufical founds be eafily explained. Let us fuppofe 
an anvil, or feveral fimilar anvils, to be ftruck upon by 
Several hammers of different weights or forces. The 
hammer, which is double that of another, upon finking 
the anvil will produce a found double that of the other: 
this double found mufieians have agreed to call an oElave. 
The ear can judge of the difference or refemblance of 
thefe founds with great eafe, the numbers being as one 
and two, and therefore very readily compared. Suppofe 
that an hammer, three times lefs than the firft, ftrikes the 
anvil, the found produced by this will be three times lefs 
than the firf't: fo that the ear, in judging the fimilitude of 
thefe founds, will find fomewhat more difficulty; becaufe 
it is not fo eafy to tell how often one is contained in three, 
as it is to tell how often it is contained in two. Again, 
fuppofe that an hammer four times lefs than the firft ftrikes 
the anvil, the ear will find greater difficulty ftill in judg¬ 
ing precifely the difference of the founds; for the diffe¬ 
rence of the numbers four and one cannot fo foon be de¬ 
termined with precifion as three and one. If the hammer 
be five times lefs, the difficulty will encreafe. If the ham¬ 
mer be fix times lefs, the difficulty ftill encreafes, and fo 
alfo of the feventh, infomuch that the ear cannot always 
readily and at once determine the precife gradation. Now 7 , 
of all comparifons, thofe which the mind makes moft eafi¬ 
ly, and with lead labour, are the moft pleafing. There 
is a certain regularity in the human foul, by which it finds 
happinefs in exa£t, ftriking, and eafily-made, comparifons. 
As the ear is but an inftrument of the mind, it is therefore 
moft pleafed with the combination of any tw r o founds, the 
differences of which it can moft readily diftinguifh. It is 
more pleafed with the concord of two founds which are to 
each other as one and two, than of two founds which are 
as one and three, or one and four, or one and five, or one 
and fix or feven. Upon this pleafure, which the mind 
takes in comparifon, all harmony depends. The variety 
of founds is infinite; but, becaufe the ear cannot com¬ 
pare two founds fo as readily to diftinguifh their difcrimi- 
nations when they exceed the proportion of obe and feven, 
mufieians have been content to confine all harmony within 
that compafs, and allowed but feven notes in mufical corn- 
polition. ' 
Let us now then fuppofe a ftringed inftrument fitted up 
5 T I C S. 87 
in the order mentioned above. For inftance, Let the firft 
firing be twice as long as the fecond; let the third firing 
be three times fhorter than the firft; let the fourth be four 
times, the fifth firing five times, and the fixth fix times, as 
fhort as the firft. Such an inftrument would probably 
give us a reprefentation of the lyre as it came firft from the 
hand of the inventor. This inftrument will give 11s all 
the feven notes following each other, in the order in which 
any two of them will accord together moft pleafingly ; but 
yet it will be a very inconvenient and a very difagreeable 
inftrument: inconvenient, for, in a compafs of feven firings 
only, the firft rnufl be feven times as long as the laft; and 
difagreeable, becaufe this firft firing will be feven times as 
loud alfo; fo that, when the tones are to be played in a dif¬ 
ferent order, loud and foft founds would be intermixed 
with the moft difgufting alternations. In order to improve 
the firft inftrument, therefore, fucceeding mufieians veiy 
judicioufly threw in all the other firings between the two 
firft, or, in other words, between the two odlaves, giving 
to each, however, the fame proportion to what it would 
have had in the firft natural inftrument. This made the 
inftrument more portable, and the founds more even and 
pleafing. They therefore difpofed the founds between the 
oftave in their natural order, and gave each its own propor¬ 
tional dimenfions. Of thefe founds, where the proportion 
between any two is moft obvious, the concord will be moff 
pleafing. Thus octaves, which are as two to one, have a moft 
harmonious effect; the fourth and fifth alfo found fweetiy 
together, and they will be found, upon calculation, to bear 
the fame proportion to each other as the odtaves do. Let 
us then ceafe to affign the coincidences of vibrations as the 
caufe of harmony, fince thefe coincidences in two firings 
vibrating at different intervals rnufl at bell be but fortui¬ 
tous; w hereas concord is always pleafing. 
But there is another property in the vibration of a mu-' 
fical firing not yet taken notice of, and which is alleged to 
confirm the foregoing theory. If we ftrike the firing of 
an harpfichord, or any other elaftic founding chord what¬ 
ever, it returns a continuing found. This till of late was 
confidered as one fimple uniform tone; but all mufieians 
now confefs, that inftead of one tone it a finally returns 
four tones, and that conflantly. The notes are, befide the 
fundamental tone, an oftave above, a twelfth above, and 
a feventeenth. One of the bafs-notes of an harpfichord 
has been diffefted in this manner by Rameau, and the ac¬ 
tual exiftence of thefe tones proved beyond a poffibility of 
being controverted. 
In the whole theory of founds, nothing has given great¬ 
er room for fpeculation, conjecture, and difappointment, 
than this amazing property in elaftic firings. The whole 
firing is univerfally acknow ledged to be in vibration in all 
its parts, yet this (ingle vibration returns no lefs than four 
different founds. They who account for the tones of 
firings by the number of their vibrations, are here at the 
greateft lofs; yet if we allow the difference of tone to 
proceed from the force, and not the frequency, of the vi¬ 
brations, this difficulty will admit of an eafy folution. 
Thefe founds, though they feem to exift together in the 
firing, aftualiy follow each other in fucceffion : while the 
vibration has greateft force, the fundamental tone is 
brought forward : the force of the vibration decaying, the 
odlave is produced, but almoft only inftantaneoufly; to this 
fucceeds, with diminifhed force, the twelfth; and, laftly, 
the feventeenth is heard to vibrate w'ith great diftinftr.efs, 
while the three other tones are always filent. Thefe founds, 
thus excited, are all of them the harmonic tones, whofe 
differences from the fundamental tone are, as was faid, 
ftrong, and diflinft. On the other hand, the difeordant 
tones cannot be heard. Their differences being but very 
fmall, they are overpowered, and in a manner drowned in 
the tones of fuperior difference: yet not always neither; 
for Daniel Bernoulli has been able, from the fame ftroke, 
to make the fame firing bring out its harmonic and its dif¬ 
eordant tones alfo. So that from hence we may juftly in¬ 
fer, that every note whatfoever is only a fucceffion of 
tones; 
