292 M U 
have been of a conftrinftion very different from thofe of 
our degenerate days, if they produced any tones that were 
ftriftly mufical. Of the millions of well-organized mor¬ 
tals who have palled by blackimiths’ (hops fince the time 
of Pythagoras, we believe no one was ever detained by a 
Jingle note, much lefs by an harmonious concord, from 
thofe Vulcanian inftruments. A different kind of noife, 
indeed, will be produced by hammers of different weights 
and fizes; but it feems not to be in the power of the mod 
fubtle ear to difcover the leaft imaginable difference with 
refpeft to gravity or acutenefs. But, though different 
noifes may be produced from different bodies, in propor¬ 
tion to their fize and folidity, and every room, chair, and 
table, in a houfe, has a particular tone, yet thefe noifes 
can never be afcertained like mufical notes, which depend 
upon reiterated and regular vibrations of the aliquot parts 
of a firing, or other elaftic body ; and, in wind-iniiru- 
ments, upon the undulation of the air conveyed into a 
tube. 
The long belief of this ftory proves that philofophers 
themfelves have fometimes taken fafts upon trull, with¬ 
out verifying them by experiment; and, as the tone of 
the hammers was afferted without proof, fo was the eft’eft 
of their different weights faftened to firings. This Ga¬ 
lileo difcovered. The numbers 6, 8, 9, 12, applied to 
different lengths of firings, would, indeed, give the in¬ 
tervals mentioned : but it is proved, that, to produce thofe 
intervals by the tenjion of different weights,the weights mull 
be the fquares of thofe numbers; that is, 3 6, 64, 81, 144. 
It is allonilhing how the blunder had been echoed from 
author to author, without experiment, till the time of 
Galileo. And Bontempi, in trying the power of weights 
upon fixings in the Pythagoric proportions of 6, 8, 9, 12, 
found, that inftead of giving the 4th, 5th, and 8th, of the 
graved tone, they produced only the minor 3d, major 3d, 
and tritonus ; fo that the whole account falls to the 
ground. But, though modern incredulity and experi¬ 
ment have robbed Pythagoras of the glory of difcovering 
• mufical ratios by accident, lie has been allowed the fuperior 
merit of arriving at.them by meditation and defign. At 
leali the invention of the harmonical canon, or mono¬ 
chord, has been afcribed to him both by ancient and mo¬ 
dern writers. 
The mufical fpeculators of Greece arranged themfelves 
into a great number of leiis, the chief of whom were the 
Pythagoreans and the Ariftoxenians. 
Pythagoras fuppoied the air to be the vehicle of found, 
and the agitation of that element, ©ccafioned by a fimilar 
agitation in the parts of the founding body, to be the 
eanje of it. The vibrations of a firing or other fonorous 
body, being communicated to the air, affeiled the audi¬ 
tory nerves with the fenfation of found ; and this found, 
he argued, was acute or grave in proportion as the vibra¬ 
tions were quick or flow. He difcovered by experiment, 
that, of two firings equal in every thing but length, the 
Ihorter made the quicker vibrations, and emitted the 
acuter found : in other words, that the number of vibra¬ 
tions made in the fame time, by two firings of different 
lengths, was inverfely as thofe lengths; that is, the 
greater the length, the fmaller the number of vibrations 
in any given time. Thus found, confidered in the vibra¬ 
tions that caufe it, and the dimenfions of the vibrating 
body, came to be reduced to quantity, and as fuch was 
the fubjeft of calculation, and expreffible by numbers. 
For inftance, the two founds that form an oftave could 
be expreffed by the numbers 1 and 2, which would re- 
prefent ejthdV’the number of vibrations in a given time, 
or the length of the firings; and would mean, that the 
acuter found vibrates twice while the graver vibrates 
once,, or that the firing producing the lower found is 
twice the length of that which gives the higher. If the 
vibrations were confidered, the higher found was as 2, 
the lower as 1 ; the reverfe, if the length was alluded to. 
In the fame manner, in the fame fenfe, the 5th would 
S I c, 
be expreffed by the ratio of 2 to 3, and the 4th by that 
of 3 to 4. 
Ariftoxenus, in oppofition to the calculations of Pytha¬ 
goras, held the ear to be the foie ftandard of mufical pro¬ 
portions. That fenle he accounted fufficiently accurate 
for mufical, though not for mathematical, purpofes; and 
it was in his opinion abfurd to aim at an artificial accu¬ 
racy in gratifying the ear beyond its own power of dif- 
tindlion. He, therefore, rejected the velocities, vibra¬ 
tions, and proportions, of Pythagoras, as foreign to the 
fubjedt, in fo far as they fubftituted abftradt caufes in the 
room of experience, and made mufic the object of in tel¬ 
ler rather than of fenfe. 
Thefe conflidting and oppofite opinions continued to 
divide the mufical and philofophical world for more than 
two thoufand years; for it was not till about the begin¬ 
ning of the lalt century that the fyftem of Pythagoras 
was confirmed by abfolute mathematical demonftration; 
fo that, now, the following propofitions, in relation to 
mufical found, have paffed from conjedture to certainty. 
1. Sound is generated by the vibrations of elaftic bo¬ 
dies, which communicate the like vibrations to the air, 
and thefe again the like to our organs of hearing. This 
is evident, becaufe founding bodies communicate tremors 
to other bodies at a difiance from them. The vibrating 
motion, for inftance, of a mufical firing, excites motion 
in others, whofe tenfion and quantity of matter difpofe 
their vibrations to keep time with the undulations of air 
propagated from the firing firft fet in motion. 
2. If the vibrations be ilbchronous, and the found mu¬ 
fical, continuing at the fame pitch, it is faid to be acuter, 
/harper, or higher, than any other found whofe vibrations 
are flower; and graver, flatter, or lower, than any other 
whofe vibrations are quicker. For, while a mufical firing 
vibrates, its vibrations become quicker by increafing its 
tenfion or diminilhing its length; its found at the fame 
time will be more acute: and, on the contrary, by dimi¬ 
nilhing its tenfion or increafin'g its length, the vibrations 
will become flower and the found graver. The like alter¬ 
ation of the pitch of the found will follow, by applying, 
by means of a weight, an equal degree of tenfion to a 
thicker or heavier and to a fmaller or lighter firing, both 
of the fame length, as in the fmaller firing the mafs of 
matter to be moved by the fame force is lei's. 
3. If feveral firings, however different in length, den- 
fity, and tenfion, vibrate altogether in equal times, their 
founds will have all one and the fame pitch, however 
they may differ in loudnefs or other qualities. They 
are called imij'ons. The vibrations of unifons are ifo- 
chronous. 
4. The vibrations of a mufical firing, whether wider or 
narrower, are nearly ifochronous. Otherwile, while the 
vibrations decreafe in breadth till they ceafe, the pitch 
of the found could not continue the fame (which we 
perceive by experience it does), unlefs where the firft vi¬ 
brations are made very violently; in which cafe, the found 
is a little acuter at the beginning than afterwards. 
Lalily. The word vibration is underliood to mean the 
time which paffes between the departure of the vibrating 
body from any afligned place and its return to the fame. 
See Phil. Tranf. for 1714. and Dr. Brook Taylor’s Me- 
thodus Incrementorum direfta et inverfa. 
After mufical ratios were difcovered and reduced to 
numbers, they were made by Pythagoras and his followers 
the type of order and juft proportion in all things: hence 
virtue, friendfhip, good government, celeftial motion, the 
human foul, and God himfelf, were harmony. This dif- 
covery gave birth to various fpecies of mulic, far more 
ftrange and inconceivable than chromatic and enharmo¬ 
nic : fuch as divine mufic, mundane mufic, elementary 
mufic, and many other divifions and fubdivifions, upon 
which Zarlino, Kprcher, and almoft all the old writers, 
never fail to expatiate with wonderful complacence. It 
is, perhaps, equally to the credit and advantage of mufic 
