Manchester Memoirs, Vo/. xlvm. (igo4), No. tS. J 



If now O A he the actual length of a canon string, draw 

 A B at right angles, meeting O P \x\ B . Cut off A B 

 equal to A B . Then A B will be an equal tempered 

 tone. From B draw in like manner B C^', and cut off 

 B C^ equal to B Cjj^. B Cjk will be another equal tempered 

 tone ; and so on. The sixth tone so described will halve 

 OA with an error of less than one part in 14,000. The 

 straight line O P may be called the "line of tones," equal 

 tempered, be it understood. 



These tones can be subdivided into semitones and 

 quarter tones of almost equal accuracy by means of the 

 following principle, well known to the Pythagoreans. 

 Let, in the following canon, 



C £> E 



O ; i ■ '■ — B 



A 



A B represent a small interval, such as a tone. Divide 

 the intercept A B into four equal parts, A C D E B. Then 

 the interval C E will be very nearly half of the interval 

 A B. Hence a line of semitones can be found by erecting 

 a perpendicular from E, equal to C E, and joining its 

 extremity with the point O ; thus the six tones can be 

 bisected ; and so likewise the semitones for quarters. 

 Testing this theorem by the tone 9 : 8, we have the ratios 

 32 : 33 : 34 : 35 : 36 ; of which we take 33 and 35. 



Now ^ X 55 _ 5 



33 33 1089 



which differs from | by one part only in 9800. In any 

 8 



case the errors of actual marking and adjusting the 



divisions of the canon far exceed the theoretical errors 



of the above sections, which, consequently, cannot be 



improved. 



The scope of the title of this paper has now been fully 



