Royal Societv. 2'jj 
poHng mi to fland in 'B-fa-h-mit which is accounted its natural 
pofition. 
Now, in order to this, Ariftoxenus and his followers took 
that of a fourth, as a known interval, by the judgment of the 
ear 5 and that of a fifth likewife : And confequently that of an 
o6lave, as the aggregate of both 5 and that of a tone as the 
difference of thofe two. 
And this of a tone, as a known interval, they took as a 
common meafure, by which they eltimated other intervals: 
Accordingly they accounted a fourth to contain two tones and 
a half; a fifth to contain three tones and a half, and confe- 
quently an eighth to contain fix tones, or five tones and two 
halftones: And at this rate our practical muficians talk of 
notes and half notes at this day; fuppofing an oi5tave to confift 
of twelve hemi-tones or half-tones 5 but Tythagoras^ and 
thofe who follow him, not taking the ear alone to be a compe- 
tent judge in a cafe fo nice, chole to diftinguifh thefe, not by 
equal intervals, but by due ratio's : And thus it was followed 
by Zarliney Kepler^ Cartes^ and others, who treat of fpecfulative 
mufick in this and the laft age; accordingly they accounted 
that of an o6lave to be, when the degree of gravity, or acute- 
ncfs of the one found to that of the other is double, or as 
2 to I : That of a fifth, Sefqui-alter^ or as 9 to 2. : That of 
a fourth, Sefqui-tertian, or as 4 to 9 ; accounting that to be 
the fweeteft ratio, which is expreffed in the fmalfefl numbers, 
and therefore next to the Untfon^ that of an OBave^ 2 to i 5 
then that of a fifth, 9 to 2; and then that of a fourth, 4. to 9 5 
and thus that of a fourth and fifth do together make an eighth, 
for f X I = 1^ r= f = 2, or the ratio of 4 to 9, compounded 
with that of 9 to 2, is the fame with that of 4 to 2, or 1 to i. 
And confequently, the difference of thofe two, which is that 
of a tone, or full note, is that of 9 to 8. For f ) f (|; or if 
out of the ratio of 9 to 2 you take that of 4 to 9, the refult is 
that of 9 to 8. 
Now according to this computation, it is manifell: that an Oc- 
tave is Ibmewhat lefs than fix full notes ; for, as was firft demon- 
flrated by Euclid^ the ratio of 9 to 8, being fix times com- 
pounded, is fomewhat more than that of 2 to i. For | x J x J 
X f X -J X f = iji^i^^ is more than fMf !| =: f . This being the 
cafe; they allowed indifputably to that uf the 'Dia-zeutic tone, 
/^, vil^ the full ratio of 9 to 8, as a thing not to be altered, be- 
ing the difference of the ^Dla-pente and Dia-tejfaron, or the 
fifth and fourth : All the difficulty was, how the remaining fourth 
