448 PHILOSOPHICAL TRANSACTIONS. [aNNO 1788. 



impracticable, because that after the number 13, if the succession of 5ths be 

 carried on further, they will recede more from a coincidence with any one of 

 the octaves. 



This explanation of the nature, origin, and necessity of the temperament has 

 been thought necessary for the sake of perspicuity; but the same end may be 

 obtained by the following easier method. As the 1 3 notes of an octave must be 

 arranged so, that whichever of them be taken for the 1st or key-note, the 2d, 

 3d, 4th, &c. may bear the same constant proportion to it; they must therefore 

 be in a geometrical proportion, so as to form a series of 1 3 numbers, the ex- 

 tremes of which are the whole string and its half, viz. any number and its half. 

 The ratio of this series is found in the same manner as in the other series, viz. the 

 greatest extreme is divided by the least, and the 1 2th root of the quotient is the 

 ratio sought. But the extremes are any assumed number and its half: and as the 

 quotient of a number divided by the half of the same number is always equal 

 to 2; therefore, whatever be the length of the string, the ratio is always 

 '^2= 1.0594 -}-; and if the length of the whole string be divided by this 

 ratio, viz. I.0594 -f-, the quotient will be the length of 

 the string expressing the 2d note, which, divided by the 

 same ratio, gives the 3d note, and so on; or else, instead of 

 dividing the length of the whole string by the ratio, we may 

 multiply the half of it by the ratio, the product of which will 

 give the 7th note, which multiplied by the same ratio gives 

 the 6th, and so on in a retrograde order, which will give the 

 tempered notes of the octaves as well as the former method. 

 By this means the annexed divisions for the notes of an octave 

 have been calculated, the length of the whole string having 

 been supposed equal to 100000. 



If a monochord be divided in this manner, and a harpsichord tuned by it, 

 this instrument will then be tuned so, that whichever note be taken for the first 

 or key-note, its 5th, 6th, &c. will produce the same effect respectively. 



At present, the harpsichords and organs are commonly tuned so, that some 

 concords are very agreeable to the ear, while others are quite intolerable; or, in 

 other words, when the performer plays in certain keys, the harmony is very 

 pleasing, in others the harmony is just tolerable, and in some other keys the 

 harmony is quite disagreeable. The best keys to be played in, are the keys of 

 c, of p, of E flat, of B flat, of g and of d in the major mood; and the keys 

 of c, of D, of A, and of b, in the minor mood. Next to those come the less 

 agreeable keys of a, a fiat, and e in the major mood; besides those, the rest 

 are disagreeable in a greater or less degree, so that out of 12 keys, which, on 



