VOL. LXXVIII.] PHILOSOPHICAL TRANSACTIONS. 443 



the vibrations which they make in a certain time. Here follow the proportions 

 which the times of vibration, or the length of the strings which express those 

 13 sounds, bear to the first, prime, or key note. 



First. 1 Fourth 4- Seventh minor -f- 



Second minor . . . . -ff- Fourth major 44 Seventh major . . . . ^ 



Second ^ Fifth -5- Octave -J- 



Third minor -|- Sixth minor -f 



Third major 4- Sixth major 4. 



If, instead of many strings having those lengths in order to express the 1 3 

 sounds, or notes of an octave, one string be divided according to those propor- 

 tions, and this string be stopped consecutively in the different points or divisions ; 

 on being struck, it will express the corresponding sounds. Thus, if a string 

 stretched between 2 fixed points, be struck, it will produce a sound called the 

 prime, first, or key-note ; if it be stopped in the middle, one half of the string 

 will sound the octave, its length, compared to that of the whole string, being in 

 the proportion of 1 to 2 ; if -§. of the string be caused to vibrate, the sound 

 produced will be the 5 th, its length, compared to that of the whole string, being 

 as 2 to 3, and so of the rest. The highest sound of the octave is expressed by 

 the half of the string ; and if this half be divided again in the same manner or 

 proportion, a higher octave will be obtained, the highest note of which will be 

 expressed by a quarter of the original string. This quarter may be divided again 

 into a higher octave, and so on ; therefore, a string so divided may express the 

 sounds of all the keys of a harpsichord or organ. 



In regard to those divisions it may be observed, that as the notes of the 2d 

 octave bear the same proportion to the first note of that octave as the notes of 

 the first octave respectively bear to the first note of that octave, or to the whole 

 string ; and as the length of the string expressing the first note of the 2d oc- 

 tave, is half the length of the first note of the first octave, it follows, that the 

 length of the string of every note in the 2d octave is half the length of the 

 corresponding note in the first octave. Hence, jvhen the divisions of the first 

 octave are ascertained, in order to find the divisions of the notes of the 2d 

 octave, we need only take the half of the lengths expressing the notes in the 

 first octave. By the very same reasoning it is evident, that to find the divisions 

 for the 3d octave, we need only take the halves of the lengths which express the 

 notes of the 2d octave, or the quarters of those of the first octave, and so of 

 the rest. 



The first string or line is divided in the above-mentioned manner, and in 

 order to avoid confusion, the divisions of the principal notes only of the first 

 and 2d octave are annexed to it. Numbers are set under the line to express the 

 lengths from the beginning to the divisions to which they stand near. The 



3l2 



