Manchester Memoirs, Vol. xlviii. (1904), No. 13. 3 



section, or marking off, of a canon string in units of 

 length ; such units being subdivisible at pleasure so long 

 as the respective ratios are maintained. It will be well to 

 divide the subject into separate propositions, as under : — 



To find the ratio of the Mean Tone. 



8081 go 



o ;:;;:;;::■■; x 



Let O X represent a canon string ; consider the 

 divisions between 80 and 90. The points give the ratio 

 8 : 9 (by dividing by their difference, which does not affect 

 the ratio) that is, the tone, or major tone. The lengths 81 

 to 90, by dividing by their difference, give the ratio 9 : 10, 

 the minor tone. Hence we can go from 90 to 8 1 , or from 

 90 to 80, so describing a minor or a major tone, with 

 theoretical accuracy. But the mean of these, the halfway 

 between the major and minor tones, would be found by 

 dividing the extra step, 80, 81 into two equal intervals; 

 this can be very nearly done by halving the intercepted 

 distance on the canon, and so calling 80^ to 90 the mean 

 tone. Doubling these numbers, to avoid the fraction, we 

 get for the ratio of the mean tone 161 : 180. I shall show 

 that one of Claudius's harmonic tetrachords enables us to 

 arrive at these very numbers in another way. 



To find the ratio of the Equal-tempered Tone. 



It has been proved by Archytas that six tones 

 (untempered) are equal to an octave and a comma of 

 Pythagoras ; therefore in order to temper, or compress, 

 six tones into the limit of an octave, each tone must be 

 curtailed by one sixth of the Pythagorean comma. The 

 ratio of this comma was accurately known ; it is very 

 nearly 73 : 74. If, then, these lengths be marked off on a 



