VOL. LXXVIII.] PHILOSOPHICAL TKANS ACTIONS. 445 



in which the notes are fixed, so as not to be alterable by the performer's hands, 

 tnust be imperfect even when tuned in the best manner possible; for by the tem- 

 perament we can divide, but not annihilate the imperfection. Other instruments, 

 in which the notes are not fixed, as the violin, violoncello, &c. are perfect, be- 

 cause the performer stops the strings on them in different places, even for sound- 

 ing the notes of the same name. Thus a skilful performer, in order to sound a, 

 will stop the string a little farther from the bridge when he plays in the key of c, 

 viz. when c is considered as the key-note, than when he plays in the key of d. 



Most people imagine, that the scale of musick is capable of many different 

 temperaments; and, agreeable to this supposition, the writers on harmonics have 

 proposed different temperaments; but the nature of the scale admits of only one 

 temperament capable of rendering the imperfection and the harmony equal 

 throughout; and it is impossible to form a different and more advantageous scale. 

 It may also be remarked, 1st, that the proportion of 2 to 3 for the fifth, the 

 proportion of 1 to 2 for the octave, and in short the proportions of all the notes, 

 are not assumed at pleasure; but have been determined from constant experience, 

 viz. from the agreeable or disagreeable effects produced when 2 different notes are 

 sounded at the same time. 



Thus, let 2 strings equal in every respect be struck at the same time, and they 

 will express the same sound precisely, so that no ear can perceive any difference 

 between them, and it is almost impossible to distinguish whether the sound 

 arises from 2 strings, or from 1 only, excepting from the loudness. But if one 

 of those strings be successively stopped in different parts of its length, while the 

 other remains open as before, and if at every time they be both struck together, 

 their combined sounds will be found to produce different effects, viz. sometimes 

 more or less pleasing, and at other times more or less disagreeable. When the 

 combinations of the 2 sounds are agreeable, they are called concords; and when 

 disagreeable, they are called discords. Experience evinces, that the best con- 

 cord is when the length of one string is to the length of the other as 1 to 2, 

 every other circumstance being the same in both. This proportion forms the 

 octave. The next best concord is the 5th, viz. when the lengths of the two 

 strings are as 2 to 3, after which cgme the proportions of 3 to 4, 4 to 5, 3 to 5, 5 to 

 6, and 5 to 8, for the other concords. The other proportions besides these are dis- 

 agreeable in a greater or less degree, unless they are greater than the proportion 

 of 1 to 2; but in that case it will be found, that the proportions which produce 

 agreeable combinations are the double, quadruple, octuple, &c. of those men- 

 tioned above, viz. are their octaves, double octaves, &c.: thus the proportion of 

 1 to 4 produces a very agreeable concord, because 1 to 4 is the double of 1 to 2, 

 viz. it expresses a double octave. 2dly, Hence if we have the length of a string, 

 or the proportion of a note in any part of the string, we may easily find its 



