VOL. XX.] PHILOSOPHICAL TRANSACTIONS. 201 



And so in each octave successively following. And if the pipes in each octave 

 be fitted to sounds in these proportions of gravity and acuteness, it will be sup- 

 posed, according to this hypothesis, to be perfectly proportioned. 



Bat, instead of these successive proportions for each hemitone, it is found 

 necessary so to order the 13 pipes, containing 12 intervals called hemitones, as 

 that their sounds, as to gravity and acuteness, be in continual proportion, each 

 to its next following, in one and the same proportion : which, all together, 

 shall complete that of an octave or diapason, as 2 to 1. Whence it happens, 

 that each pipe does not express its proper sound, but very near it, yet some- 

 what varying from it ; called bearing. Which is somewhat of imperfection in 

 this noble instrument, the chief of all. 



It may be asked. Why may not the pipes be so ordered, as to have their 

 sounds in just proportion, as well as thus bearing .'' I answer, it might very well 

 be so, if all music were composed to the same key, or, as the Greeks call it, 

 the same mode. As for instance, if, in all compositions, mi were always placed 

 in b fa b mi. For then the pipes might be ordered in such proportions as I 

 have now designed. But musical compositions are made in great variety of 

 modes, or with great diversity in the pitch. Mi is not always placed in b fa 

 b mi ; but sometimes in e la-mi ; sometimes in a la-mi-re, &c. And in some there 

 is none of these 12 or 13 pipes, but may be made the seat of mi. And if they 

 were exactly fitted to any one of these cases, they would be quite out of order 

 for all the rest. As for instance ; if mi be removed from b fa b mi, by a fiat 

 in B, to E la-mi: instead of the proportions before designed, they must be thus 

 ordered ; 



Where it is manifest, that the removal of mi quite disorders the whole series 

 of proportions. And the same would again happen, if mi be removed from e 

 to A, by another flat in e. And again, if removed from a to d. And so perpe- 

 tually. But the hemitones being made all equal ; they indifferently answer all 

 the positions of mi, though not exactly to any : yet nearer to some than to 

 others. Whence it is, that the same tune sounds better at one key, than at 

 another. 



It is asked. Whether this may not be remedied, by interposing more pipes; 

 and thereby dividing a note, not only into half notes but into quarter notes 

 or half quarter notes, &c. I answer : it may be thus remedied in part ; the im- 

 perfection might be somewhat less, and the sounds somewhat nearer to the just 



p p 2 



