VOL. XX.] PHILOSOPHICAL TRANSACTIONS. 241 



greater exactness is not thought necessary. And this is very convenient, because 

 thus the change of the key, upon altering the place of mi, gives no new trou- 

 ble ; for this indifferently serves any key, and the difference is so small as not 

 to offend the ear. 



But those who choose to treat of it with more exactness, proceed thus : pre- 

 supposing the ratio for an octave, or diapason, to be that of 2 to 1 ; they divide 

 this into two proportions; not just equal, for that would fall on surd numbers, 

 as of y/1 to 1 ; but nearly equal, so as to be expressed in small numbers. In 

 order to which, instead of taking 2 to 1, they take the double of these numbers 

 4 to 2 ; which is the same ratio as before, and interpose the middle number 3. 

 And of these three numbers 4, 3, 2, that of 4 to 3 is the ratio of a fourth, or 

 diatesseron ; and that of 3 to 2, the ratio of a fifth, or diapente ; and these two 

 put together make up that of an octave, or diapason, that of 4 to 2, or 2 to 1. 

 And the difference of those two that of a tone or Q to 8 ; as will plainly appear 

 in the ordinary method of multiplying and dividing fractions. That is, \ y, \ 

 = A = i. And A ) 4 ( :^. 



Thus, in the common scale, or gamut, taking an octave in these notes, 

 la, fa, sol la, mi, fa, sol, la ; (placing mi in b fa tl mi, which is called the 

 natural scale) the lengths for the extremes la, la, an octave, are as 2 to 1, 

 or 12 to 6 ; those for la, la, (in la fa sol la) or mi la, (in mi fa sol la) a 

 fourth, as 4 to 3, or 12 to Q, or 8 to 6 ; those for la mi (in la fa sol la mi) or 

 la la (in la mi fa sol la) a fifth, as 3 to 2, or J 2 to 8, or 9 to 6 ; those for la mi, 

 the diazeuctic tone or difference of a fourth and fifth, as Q to 8. Thus we have 

 for these four notes, la, la, mi, la, their proportional lengths in the numbers 

 12, 9, 8, 6. 



Then if we proceed in like manner to divide a fifth, or diapente, la, fa, sol, 

 la, mi, or la, mi, fa, sol, la, or the ratio of 3 to 2, into near equals, taking 

 double numbers in the same ratio, 6 to 4, and interposing the middle number 5 ; 

 of these three numbers 6, 5, 4, that of 6 to 5 is the ratio of a lesser third, 

 called a trihemitone, or tone and half, as la, fa, (in la, mi, fa,) ; and that. of 5 

 to 4, is the ratio of the greater third, commonly called a ditone, or two tones, 

 as fa, la, (in fa sol la) which two put together make a fifth, as 3 to 2 ; that is 

 -I X f = -i- = I- ; a'ld their difference is as 25 to 24, that is f ) f ( -|4-. So have 

 we for these 3 notes la, fa, la, their proportionate lengths in numbers, as 6, 5, 4. 

 In like manner if we divide a ditone or greater third, as fa, la, (in fa, sol, la) 

 whose ratio is as 5 to 4, or 10 to 8, into two near equals, by help of a middle 

 number 9 ; then we have in these three numbers 10, 9, 8, that of 10 to 9, for 

 what is called the lesser tone, and that of 9 to 8 for the greater tone. But 

 whether fa sol shall be made the lesser, as 10 to 9, and sol la the greater, as 



VOL. IV. I I 



