2Q2 PHILOSOPHICAL TRANSACTIONS. [anNO ifiQb. 



proportions ; but it can never be exactly true, so long as their sounds, be they 

 ever so many, are in continual proportion, that is, each to the next following 

 in the same proportion. 



For it has been long demonstrated, that there is no such thing as a just hemi- 

 tone practicable in music. For, supposing the proportion of a tone or full 

 note to be -§-, or as 9 to 8 ; that of the half note must be as y/g to \/6, as the 

 square- root of 9 to the square-root of 8, that is, as 3 to \/8. or 3 to "2 v' 2, 

 which are incommensurable quantities. And that of a quarter note, as 1/9 to 

 ^8, as the biquadrate root of 9 to the biquadrate root of 8, which is yet more 

 incommensurate. And the like for any other number of equal parts. Which 

 will therefore never fall in with the proportions of number to number. So that 

 this can never be perfectly adjusted for all keys, without somewhat of bear- 

 ing, by multiplying of pipes, unless we would for every key, or every different 

 seat of mi, have a different set of pipes, of which this or that is to be used, 

 according as, in the composition, mi is supposed to stand in this or that seat. 

 Which vast number of pipes for every octave would vastly increase the charge. 

 And, when all is done, make the whole impracticable. Those who desire to 

 know more of it, may see my thoughts more at large in the appendix to my 

 edition of Ptolemy's Harmonics, in Greek and Latin. 



The two eminent sects among the ancients, the Aristoxenian and the Pytha- 

 gorean, differ much in the same way as the language of our ordinary practical 

 musicians, and that of those who treat of it in a more speculative way. Our 

 practical musicians talk of notes and half notes, just as the Aristoxenians did, 

 as if the whole notes were all equal, and the half notes likewise each the just 

 half of a whole note. But Pythagoras, and those who follow him, found, by 

 the ear, that this equality of intervals would not exactly answer the musical 

 appearances in concords and discords, just as our organists and organ-makers 

 are now aware, that their pipes at equal intervals do not give the just desired 

 harmony, without somewhat of bearing, that is, of some little variation from 

 the just sound. The Pythagoreans, to help this, changed the notion of equal 

 intervals into that of due proportions ; and this is followed by Zarline, Kepler, 

 Cartes, and others who treat of speculative music in this and the last age. And 

 though they speak of notes and half notes, in a more gross way, much as others 

 do, yet they declare themselves to be understood more nicely. 



And though our present gamut take no notice of this little diversity, yet, in 

 vocal music, the ear directs the voice to a more just proportion. And, in string 

 music, it may in like manner be helped by straining and slackening the strings, 

 or meving the frets. But, in wind music, the pipes are not capable of such 



