Royal Society. 245 
fourth, or ^Diateffarotu the felqui-tertian, or that of 4 to 3 • and 
to a tone, which is the difference of a fourth and fifth, the lel- 
qui-o6tave, or that of 9 to 8; becanlc lengths, taken in the fame 
firing in thefe ratios, do give fuch Ibunds: And univerfally, 
whatever ratio of lengths, taken in the fame firing equally 
ftretched, gives iuch and fuch Ibunds, juft fuch ratio's of gravity 
we affign to the founds fo given : But when an eight or oflave is 
faid in common fpeech to confifl of 12 hemi-tones, or 6" tones, 
this is not to be underllood according to the utmoft rigour of ma- 
thematical exa6lnefs, of fuch 6 tones, as are called tfie 2Jiazeu- 
tic tone, or that of la^ mi, which is the difference of a fourth 
and fifth, but as fufliciently exa6l for common uie^ for 6 fuch 
tones, that is, the ratio of 9 to 8 repeated fix times, are ibme- 
what more than an oclave, or the ratio of 2 to 1 5 and confe- 
quently fuch an hemi-tone is fomewhat more than the twelfth part 
of an eight, or o6lave, or 2)lapafon'j but the ditterence is fo 
fmall, that the ear can hardly diftinguifh it 5 and therefore in 
common fpeech it is ufual ib to fpeak : And accordingly, when 
we are dire^ed to take the lengths for what are called the 12 
hemi-tones in geometrical proportion, it is to be underflood not 
to be fb in the utmoft flridlnefs, but to be accurate enough for 
common ufe, as for placing the frets on the neck of a viol, or 
other mufical inftruraent, wherein a greater exaflnels is not 
thought neceffary 5 and this is very convenient, becaule thus the 
change of the key, upon altering the place of ?;;/, gives no new 
trouble 3 for this does indifferently ferve any key, and the diflf^- 
rence is io fmall, as not to offend the ear: But fuch as chufe to 
treat of it with more exadnefs proceed thus- prefuppofing the 
ratio of an o£lave or 2)iapafon to be that of 2 to i, they divide 
this into two ratios, not juft equal, for that would fall on the 
furd numbers, as V^ to i ; but nearly equal, fb as to becxprefled 
in fmall numbers 5 in order to which, inftead of taking 2 to i, 
they take the double of thefe numbers, vi-z,. 4 to 2, which is the 
fame ratio as before, and interpofe the middle number 5 5 and of 
thefe three numbers, 4, 5, 2, that of 4 to 5 is the ratio of a 
fourth, or Uiatejjaron 3 and that of 5 to 2, the ratio of a fifth, 
or Diapente •-, and thefe two, put together, make up that of an 
oflave, or ijiapaforiy that of^ 4 to 2, or 2 to i • and the 
difference of thefe two, that of a tone, or 9 to 8, as will 
plainly appear by the ordinary method of multiplying and di- 
viding fraaions, that is, f x'f = ^ r=: f 3 and f) i ( f = Thus 
in the common fcale, or gamut, taking an odlave in thefe notes 
la J fa^ foh hi ml) fa^fol) la, fuppofe from E to ^ (placing rait 
