288 PHILOSOPHICAL TRANSACTIONS. [aNNO 16Q9. 



and sharpness, is the ground of what we call concord and discord ; that is, a 

 soft or harsh coincidence. 



Now, concerning this, there were among the ancient Greeks, two of the 

 most considerable sects of musicians : the Aristoxenians, and the Pythagoreans. 

 They both agreed thus far ; that Diatessaron and Diapente do together make 

 up Diapason ; that is, as we now speak, a fourth and fifth do together make 

 an eighth or octave.: and the difference of those two, viz. of a fourth and fifth, 

 they agreed to call a tone ; which is now called a whole note. 



Such is that, in our present music, of la-mi, or as it was wont to be called, 

 re-mi; for la- fa-sol-la, or mi-fa-sol-la, is a perfect fourth: and la-fa-sol-la-mi, 

 or la-mi-fa-sol-la, is a perfect fifth : the difference of which, is la-mi. 'Which 

 is, what the Greeks call the diazeuctic tone ; which disjoins two fourths, on 

 each side of it ; and, being added to either of them, makes a fifth. Which 

 was, in their music, that from mese to paramese ; that is, in our music, from 

 A to B : supposing mi to stand in b fa b mi, which is accounted its natural 

 position. 



Now, in order to this, Aristoxenus and his followers took that of a fourth, 

 as a known interval, by the judgment of the ear; and, that of a fifth, likewise; 

 and consequently, that of an octave, as the aggregate of both ; and of that a 

 tone, as the difference of those two. 



And this of a tone, as a known interval, they took as a common measure, 

 by which they estimated other intervals. And accordingly they accounted a 

 fourth to contain two tones and a half; a fifth to contain three tones and a 

 half; and consequently an eighth to contain six tones, or five tones and two 

 half-tones. And it is very near the matter, though not exactly so. And at 

 this rate we commonly speak at this day ; supposing an octave to consist of 12 

 hemitones, or half-notes, meaning thereby somewhat near so many half-notes : 

 but when we would speak more nicely, we do not take those supposed 

 half-notes to be exactly equal, or each of them just the half of a full-note, 

 such as la-mi. 



Pythagoras, and those who follow him, not taking the ear alone to be a 

 competent judge in a case so nice, chose to distinguish these, not by intervals, 

 but by proportions. And accordingly they accounted that of an octave, to be, 

 when the degree of gravity or acuteness of the one sound to that of the other 

 is double, or as 2 to 1 ; that of a fifth, when it is sesqui-alter, or as 3 to 2 ; 

 that of a fourth when sesqui-tertian, or as 4 to 3. Accounting that the sweetest 

 proportion, which is expressed in the smallest numbers ; and therefore, next to 

 the unison, that of an octave, 2 to 1 ; then that of a fifth, 3 to 2 ; and then 



