VOL. XXIV.] PHILOSOPHICAL TRANSACTIONS. 245 



sounds are required for expressing those seven gradual steps, by which we com- 

 monly ascend to it. It may be also observed, that the proportions falling upon 

 the same notes in two keys, one finger-board will be sufficient for both. 



It is acknowledged by all who are acquainted either with speculative or prac- 

 tical music, that every interval is divided into two parts, one of which is greater 

 than the other: an eighth ±, into a fifth 4, and a fourth -|-. Again, a fifth 4, 

 into a greater third -*-, and a lesser third f. Thus also a greater third a must 

 be divided into a tone major -§-, and a tone minor -^-g-. The lesser third (to 

 comply with the practice of music) is rather compounded of, than divided into 

 a tone major -f-, and a hemitone, which is its complement, ■\-^. 



Three tones major, two tones minor, and two of the foresaid hemitones, 

 placed in the order found in the scheme, exactly constitute the practical octave; 

 which is so called, because it consists of eight sounds, that contain the seven 

 gradual intervals. But it is also necessary to set down the divisions of the 

 whole tones, which are the true chromatic half notes, because there is great 

 use of them in practical music. 



To make all our whole notes, and all our half notes of an equal size, by 

 falsifying the proportions, and bearing with their imperfections, as the common 

 practice is, may be allowed by such ears as are vitiated by long custom: but it 

 certainly deprives us of that satisfactory pleasure which arises from the exact- 

 ness of sonorous numbers; which we should enjoy, if all the notes were truly 

 given according to the proportions here assigned. 



It is very easy to satisfy ourselv^es in the arithmetical scheme, by those ope- 

 rations which Gassendus has set down in his Manuduction to the theory 

 of music, tom. v, p. 635. As for example, his rule for addition is, that two 

 proportions being given, if the greater number of one be multiplied by the 

 greater number of the other, and the lesser by the lesser, the two numbers 

 produced exhibit the compounded proportions. Thus, take a practical fifth •§-, 

 and a practical fourth -f-, for the two proportions given; multiply 3 by 4, and 

 you have 12 ; then multiply 2 by 3, and you^ave 6: which compounded pro- 

 portion of 12 to 6, makes the practical octave ±. Thus, according to his 

 arithmetical operations of addition, subtraction, multiplication or continuation, 

 and division, is our whole system proved; which, for the more easy application 

 to practical music, may be exhibited geometrically on 6 parallel lines, represent- 

 ing the 6 strings of a viol. 



This mathematical fixing of the frets enables every practitipner, who stops 

 close to them, to give the proportions of the notes in a greater exactness, than 

 can be done on the bass-violin or violin itself; since they may be set forth 



