Manchester Memoirs, Vol. xlviii. (1904), No. 13. 5 



In the Book mentioned, Claudius develops a system of 

 tetrachords of his own invention, and adds the teniae of 

 Archytas, thus giving three diatonic, two chromatic, and 

 one harmonic tetrachord. It is this last only which here 

 concerns us. His reasons for the section are a mystery to 

 Dr. Wallis, his translator ; his method is as follows : — 



After dividing the ratio of the fourth, 4 : 3, into the 



two, 5 :4 and 16: 15, he triples the last numbers, giving 



45, 46, 47, 48. He rejects 47, because it does not yield a 



" superparticular" ratio with the extremes, and so selects 



45, 46, 48, or — — for the division of 16: 15 ; placing 



45' ^3> 



the smaller interval at the grave extremity. Thus the 

 complete section is denoted by the tetrachordal formula : — 



4^x^x5 = 4 

 45 23 4 3 



Now one prominent member of his diatonic tetra- 

 chords is the ratio 8 : 7, which is about a minor tone and 

 a quarter ; let us examine the effect of curtailing it by the 

 interval 46 :45. We have at once : — 



8^45=12145 = 1^ 

 7 46 7 X 23 161' 



the numbers already obtained for the mean tone. Hence 

 follows the construction : — 



M N TX 



o—. ; \ — w- 



14 15 16 



Let 6^ Jf be a straight line, not necessarily a canon 

 string, divided into 16 equal parts, of which it is only 

 necessary to consider M, N and X, situate at 14, 15, 16 

 severally. Then MX corresponds to the ratio 16:14, or 

 8:7; i\^Z to 16:15. 



N X has now to be divided in the point T, such that 



