VOL. XX.] PHILOSOPHICAL TRANSACTIONS. Q.^Q 



that of a fourth, 4 to 3. And thus, that of a fourth and fifth, together make 

 an eighth ; for -t X -I = -^ = -r = 2. That is, four-thirds of three halves, is 

 the same as four halves, that is 2. Or the proportion of 4 to 3, compounded 

 with that of 3 to 2, is the same with that of 4 to 2, or 2 to i. And, con- 

 sequently, the difference of those two, which is that of a tone or full-note, is 

 that of 9 to 8. For a) |- (| ; that is, three halves divided by four-thirds, is 

 mine eighths ; or, if out of the proportion of 3 to 2, we take that of 4 to 3, 

 the result is that of 9 to 8. 



Now, according to this computation, it is manifest, that an octave is some- 

 what less than 6 full-notes. For the proportion of 9 to 8, being 6 times com- 

 pounded, is somewhat more than that of 2 to 1. For |--|--S--(-|.-fj._|-|.-]-»_ 

 =: 4|jj»44, is more than 4|-^_f_|-|- := «-. This being the case; they allowed in- 

 disputably to that of the diazeuctic tone, la-mi, the full proportion of 9 to 8 ; 

 as a thing not to be altered ; being the difference of diapente and diatessaron, 

 or the fifth and fourth. All the difficulty was, how the remaining fourth, 

 mi-fa-sol-la, should be divided into three parts, so as to answer pretty near the 

 Aristoxenians two tones and a half; and might altogether make up the propor- 

 tion of 4 to 3 ; which is that of a fourth or diatessaron. 



Many attempts were made to this purpose ; and according to those, they 

 gave names to the different genera or kinds of music, the diatonic, chromatic, 

 and enarmonic kinds, with the several species, or lesser distinctions under those 

 generals. The first was that of Euclid, which most generally obtained for many 

 ages : it allows to fa-sol, and to sol-la, the full proportion of 9 to 8 ; and there- 

 fore to fa-sol-la, called the greater third, that of 81 to 64. For f X -f- = f^J-. 

 And, consequently, to that of mi-fa, which is the remainder to a fourth, that 

 of 256 to 243. For -f-i-) 4- (i-Af ; that is, if out of the proportion of 4 to 3, 

 we take that of 81 to 64, the result is that of 25(5 to 243. To this they gave 

 the name of limma (Atrjuf/,*), that is, the remainder, viz. over and above 2 tones. 

 But in common discourse, when we do not pretend to speak nicely, nor intend 

 to be so understood, it is usual to call it a hemitone or half-note, being very 

 near it, and the other, whole notes. And this is what Ptolemy calls diatonum 

 ditonum, of the diatonic kind with 2 full tones. 



Against this, it is objected, as not the most convenient division, that the 

 numbers of 81 to 64, are too great for that of a ditone or greater third ; which 

 is not harsh to the ear ; but is rather sweeter than that of a single tone, the 

 proportion of which is 9 to 8. And in that of 256 to 243, the numbers are 

 yet much greater. Whereas there are many proportions, as a, a, ^, -f, in 

 smaller numbers than that of 9 to 8 ; of which, in this division, no notice 

 is taken. 



VOL. IV. P T 



