444 PHILOSOPHICAL TRANSACTIONS, [annO 1788. 



letters just over the lines are the names of the notes or sounds expressed by the 

 corresponding lengths of the string. The fractional numbers express the pro- 

 portion which each particular division bears to the whole string; and the Roman 

 numbers denote the numerical names of each note with respect to its distance 

 from the first, which is always included. It is evident, that if any of those di- 

 visions be considered as the first or key-note, then the other notes, though they 

 retain their alphabetical names, must have their numerical names altered ac- 

 cordingly: for example, if we take d for the key-note, then a will be the 5th of 

 it, whereas a was the 6th when c was considered as the key-note; thus also b is 

 the 3d of G, and the 7 th of c; and so on. 



Thus much having been premised, we may proceed to show the meaning of 

 what is called the temperament in a system of musical sounds, and the necessity 

 of it. For this purpose it is necessary to recollect, first, that the string, divi- 

 ded in the above-mentioned manner, exhibits the various notes or sounds of the 

 keys of a harpsichord, the pipes of an organ, &c. 2dly, That those divisions 

 remain unalterable, so that the harpsichord, when tuned, cannot be altered in, 

 the course of performing on it. And, 3dly, that when any of those notes or 

 divisions is considered as the key-note, its 2d, 3d, 4th, 5th, &c. must bear 

 their respective proportions, according to what has been said above. Now if, 

 among the divisions of the first string cz, we take d for the first or key-note, its 

 length being 320 inches, the length of its 5th must be 213-1- inches, viz. -§- of 

 320, that being the proportion which the 5th must bear to the key-note; but 

 among the divisions of the string, there is none equal to 213^ inches; there- 

 fore, there is not a note among them which may serve for a 5th to d; however, 

 as the length of az, viz. 21 6, is the nearest to 21 Si, this a must be taken for 

 the 5th of D. It is evident, that this is an imperfect 5th of d; but if, in order 

 to render it perfect, we make az equal to 2134- inches instead of 21 6, then it 

 will be a redundant 6th to c, when c is considered as the key-note; the best 

 expedient therefore, is to divide the imperfection between the 2 lengths, viz. to 

 make az neither so long as 21 6, nor so short as 213J-, which will render the dis- 

 agreeable sensation, arising from the improper length, the least possible. This 

 alteration of the just lengths of strings, necessary for adapting them to several 

 key-notes, is called the temperament: and the best temperament in a set of mu- 

 sical sounds is evidently such a partition of the natural imperfections, as will 

 render all the chords equally and the least disagreeable possible. 



What has been exemplified in d and A may be said of all the other notes; so 

 that if any one of them be a perfect 3d, 5th, &c. with respect to one key-note, 

 it will be found to be imperfect with respect to others. Hence it is manifest, 1st, 

 that in a set of musical keys, pipes, or frets, a temperament is absolutely ne- 

 cessary ; and, 2dly, that the harpsichord, organ, guitar, or any other instrument 



